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Q: You guys talk a lot about real analysis, limits, and calculus; shouldn't this just be about arithmetic?

A: Unfortunately, in order to formally prove many qualities of numbers, one often has to resort to higher mathematics: real analysis in the case of real numbers, number theory in the case of integers, and so forth. The article and arguments page both aim to be understandable to all, but, since many skeptics ask for formal proofs, higher mathematics will inevitably come into play.

Q: Person X made a pretty convincing argument that ${\displaystyle 0.999...\neq 1}$! Those who say otherwise are giving arguments I can't understand.

A: Before believing an argument, check sources, responses, and record. The main article is well-sourced, whereas arguments against generally cite (if anything) non-mathematical sources such as online message boards and dictionaries. Also, although most people are trying to write something everyone can understand, some arguments, in relying on higher mathematics, will not be easy to follow for all. Still, try to follow those you can. Finally, those who firmly believe that mainstream mathematics is mistaken will generally reveal their lack of rigor and/or contempt for experts and others who disagree with them. Before replying, read their other contributions to make sure you aren't siding with someone you yourself would not trust.

Q: I have a mathematical question.

A: Please check the FAQ on the talk page.

Q: Do any reliable sources side with the students' stubborn feeling that 0.999... should be less than 1?

A: Yes. See the section on infinitesimals in the main article.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10
11, 12

Comments of/additions to?, Formal proof

The in "Formal proof" given additon rule for decimal numbers implies

${\displaystyle 0.(9)_{n}+1/10^{n}=1,}$

But than

${\displaystyle 0.(9)_{n}-1=1/10^{n},}$

and if

${\displaystyle 0.(9)_{n}=1}$

than ${\displaystyle 1/10^{n}}$ must equal 0. In that case either 1 or ${\displaystyle 10^{n}}$ must be equal to 0. But we know that 1 as well as ${\displaystyle 10^{n}}$ is > 0. Ergo: ${\displaystyle 1/10^{n}}$ must be for every ${\displaystyle n>0}$ and thus ${\displaystyle 0.999...<1}$.

Euclid's reductio ad absurdum proved that assuming a possibly largest prime number was an absurdity, <ref> https://en.wikipedia.org/Largest_known_prime_number<ref>.

This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer.

But since all positive prime numbers are odd numbers, it follows from Euclid’s proof that the assumption of a possibly largest positive integer ${\displaystyle n}$, either even or odd, must be absurd too. So for every ${\displaystyle n}$ there must be a number ${\displaystyle n+n_{1}}$, ${\displaystyle n_{1}}$ being a number ≥ 1, that is (much) larger than ${\displaystyle n}$. From this follows that ${\displaystyle 1/10E(n+n_{1})}$ < ${\displaystyle 1/10^{n}}$ > 0. ˜˜˜˜

You're assuming that if something s true for every natural number it is true at the limit. This is like saying all natural numbers are finite, therefore infinity does not exist. The correct conclusion is we can only talk about the finite numbers unless we have an extra axiom.. In this case we can't say anything about the limit without the Archimedean property or completeness axiom as discussed near the start of the article. Dmcq (talk) 12:35, 2 December 2017 (UTC)

Dear madam/sir, Thank you for your comment. You write: “You're assuming that if something s true for every natural number it is true at the limit.” I’m sorry, but no, I’m not asssuming that. Using Euclid’s Reductio I contend that it is impossible that there could be a limit to the infinity of possible natural numbers. In my opinion logically, I conclude that for every natural number ${\displaystyle n}$ one can create or think of, there must be a number ${\displaystyle n+n_{1}}$, ${\displaystyle n_{1}}$ ≥ 1. Where do you think lies the error in my logic. Can you specify a limit with regard to the infinite multitude of natural numbers? Or can you prove that Euclid ‘s Reductio is not 'true'? Antonboat (talk) 11:56, 3 December 2017 (UTC)

It is perfectly true that 0.999...999 with any natural number of nines denotes a real number that is less than 1. But "0.999..." is shorthand for saying that there are an infinite number of nines. Infinity is not a natural number. So while you have indeed proved that 0.9, 0.99, 0.999, and so on are all less than 1 with any arbitrary natural number of nines, it does not follow that 0.999..., which has an infinite number of nines, is less than 1. In fact it is not.

If you are just uncomfortable with the idea of an infinity that is larger than any of the naturals, there is another approach. 0.999... can be defined as the limit of that sequence (0.9, 0.99, 0.999, ...), i.e. a number separate from the sequence which these terms get as close as you want to (but might not reach). We can also see that as you go further and further down that sequence you will hit terms that are as close to 1 as you like: if you would like a term that is no further than 10 from 1, take the (n+1)st term. And the sequence doesn't fool around: once the terms get within this distance of 1, they don't move back away, and as you get further and further in the sequence the distance to 1 gets below any nonzero real number you want, without reaching zero, of course. So the limit is equal to 1.

We also defined 0.999... to mean this limit, so 0.999... = 1. Again, the limit is not a member of the sequence, so just because you have correctly proved that every member of the sequence is less than 1 does not preclude its limit from being 1. Double sharp (talk) 12:52, 3 December 2017 (UTC)

Dear Double sharp. You write: “ But "${\displaystyle 0.999...}$" is shorthand for saying that there are an infinite number of nines. Infinity is not a natural number.” Euclid showed us that it is impossible to express the infinite multitude of nines in a natural number because there is no limit to this multitude. Please send me an concrete example of the transition from a natural number to a non-natural infinite number.

You write: “If you are just uncomfortable with the idea of an infinity that is larger than any of the naturals, there is another approach. ${\displaystyle 0.999...}$ can be defined as the limit of that sequence (${\displaystyle 0.9,0.99,0.999,...}$), i.e. a number separate from the sequence which these terms get as close as you want to (but might not reach).” I do not agree with you, your: “(but might not reach)”, must, in my opinion because of Euclid’s Reductio, read: ”(but shall never reach)” And because of this in my opinion ${\displaystyle 0.999...}$ must be interpreted as an infinite sequence of real numbers ${\displaystyle <1}$.Antonboat (talk) 16:24, 4 December 2017 (UTC)

Some sequences reach their limit. (1, 1, 1, 1, ...) certainly does, for example. It's just that this one does not.

Again, 0.999... doesn't represent the sequence: it represents the limit of the sequence. It's not part of the sequence (0.9, 0.99, 0.999, ...): it's specifically defined to be the number that these terms approach, which they happen not to reach. So saying that 0.999... = 1 doesn't mean that we have to have some number with a finite number of nines that is equal to 1. Double sharp (talk) 06:05, 10 December 2017 (UTC)

You seem to be asking for a proof of the existence of an actual infinity. We can't do that - it all depends on having an extra axiom as I said before. But I can indicate the existence with a simple example. For every natural number N we can associate an apple with each number up to N - but we can't redistribute the apples so there is no apple associated with one of the numbers and all the other numbers have one apple at most associated. However with all the natural numbers we could redistribute the apples so the apple that was associated with number n is now associated with 2n. This means there is now just one apple associated with each even number and none with the odd numbers. This means that even though something is true for every particular N it is not true for all the numbers. This is an example of N going to the limit and something not then being true. Dmcq (talk) 13:04, 3 December 2017 (UTC)

Dear Dmcq. You write: “You seem to be asking for a proof of the existence of an actual infinity. We can't do that - it all depends on having an extra axiom as I said before.” And indeed I am asking for such a proof. And I knew beforehand that you could not produce that. Because Euclid’s Reductio showed us all that the assumption of an actual infinity concerning what ever, be it numbers, time or lines is an absurdity. That’s why infinity cannot be a number. Therefore an axiom: “there is an actual infinity”, must be contrary to Euclid’s Reductio.

You conclude your example: “This means that even though something is true for every particular N it is not true for all the numbers. This is an example of N going to the limit and something not then being true.” But concerning the whole numbers Euclid’s Reductio implies that the infinite multiplicity of natural numbers has no limit, therefore one can not speak of "all numbers", unless ... one starts from an (according to Euclid’s Reductio) absurd axiom: “There is an actual infinity”.Antonboat (talk) 16:24, 4 December 2017 (UTC)

Yes, Antonboat, indeed, as you say, "every number in the infinte sequence ${\displaystyle 0.9,0.99,0.999,...}$ is smaller than 1", but we can prove that we can "get as close to 1 as we like", in the following sense: you give me any positive distance you want, and I can give you a number in the sequence that is closer to 1 than your chosen distance. (Go ahead, try it, give me a distance of your choice.) That can be proven, and that is abbreviated as ${\displaystyle 0.999...=1}$. It is that simple, and the article says it at the end of section 0.999...#Infinite series and sequences. That is the only section in the entire article that matters. You can safely forget everything else. - DVdm (talk) 16:45, 4 December 2017 (UTC)Antonboat (talk) 18:56, 4 December 2017 (UTC)

Dear Dvdm. I am sure that you can give me a number in the sequence that is closer to 1 for every distance I should choose. Because, as I wrote in my original note, this follows from ${\displaystyle 1/10E(n+n_{1})}$ < ${\displaystyle 1/10^{n}}$ > ${\displaystyle 0}$, ${\displaystyle n_{1}}$ being ≥ 1, for any ${\displaystyle n}$ one has chosen. But can you give me a real number ${\displaystyle r}$ in the aforementioned sequence for which ${\displaystyle r}$ = 1?

No, of course not. There would be no reason to give such a number. - DVdm (talk) 19:13, 4 December 2017 (UTC)

Just saying Reductio doesn't show anything if you don't actually cover the various options. And you haven't. Without some extra axiom .99999... does not have a particular value. With the Archimedean axiom it is 1. There are other systems in which infinitely small numbers exist so it is infinitesimally less than 1, but I get the feeling you'd find that even less palatable, the standard number system does not have numbers which are different but cannot be distinguished by a sufficiently good ruler. Dmcq (talk) 17:22, 4 December 2017 (UTC)

Dear Dmcq. You’re right as you say: “Just saying Reductio doesn't show anything if you don't actually cover the various options. And you haven't.“ In the future I’ll try not to do it again. But please correct me in that respect if I fail and my argument becomes unclear. You wrote: “Without some extra axiom ${\displaystyle .99999...}$ does not have a particular value.” I agree. I try to gain more insight into the mathematical view of the infinitely small and (actual) infinite. In that context, I consider the assumption of infinitesimals > 0 as not in conflict with the expression: ${\displaystyle 0.999...}$ < 1 and with Euclid's in the original note extensively explained Reductio.Antonboat (talk) 18:56, 4 December 2017 (UTC)

Dear Dmcq. You write: “Without some extra axiom ${\displaystyle .99999...}$ does not have a particular value. With the Archimedean axiom it is 1.” In: https://en.wikipedia.org/Axiom is said: “As used in mathematics ... an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived.“ Starting from the Archimedean Axiom you argue that ${\displaystyle 99999...=1}$. Starting from Euclids Reductio I argue that ${\displaystyle 99999...<1}$. Could it be possible that we both be right?Antonboat (talk) 14:39, 5 December 2017 (UTC)

Just because you think something is absurd does not mean that it is absurd or wrong. Reductio ad absurdum is simply the principle that a supposition that proves a falsity must be wrong, and it predates Euclid. You have not logically proved anything at all. The article covers the various options. In the standard model of the real numbers the value is 1. In some other systems infinitesimals exist. The axioms are different for the different systems. See the section 'In alternative number systems' in the article. Dmcq (talk) 18:22, 5 December 2017 (UTC)

Dear Dmcq,The reason I did not respond at once to your comment is that I did not get a signal that there were any responses. Reading your comment I realize that instead of writing “Reductio ad Absurdum” I better should have written: “Euclid's Theorem” https://en.wikipedia.org/Euclid%27s_theorem. You write: “Just because you think something is absurd does not mean that it is absurd or wrong.”

First a linguistic aside. As my native language is Dutch we say “absurd” or: “ongerijmd”. For this last word Google translate advises the translation: ¨absurd”. In Dutch the word “ongerijmd” means :“te onwaarschijnlijk om waar te kunnen zijn”, in translation: “too unlikely to be true” . I think this fits best with the word “illogical”: according to Google meaning: “lacking sense or clear, sound reasoning.”

I wrote that assuming an actual infinity, speaking of: all the real numbers, etc is absurd. (But from now on I‘l write “illogical” instead of “absurd”.) I wrote this on the authority of Euclid’s proof. This proof convinced me that neither a largest prime number, nor a largest natural number can exist. So, once again, see: 11:56, 3 December 2017, can you prove that Euclid ‘s Theorem is not 'true'?Antonboat (talk) 10:42, 11 December 2017 (UTC)

Take the decimal expansion 0.999..., with one 9 for each decimal place. Now we can index those decimal places as {1, 2, 3, ...}, corresponding to the natural numbers: one 9 for each natural number. But in that case no natural number suffices to represent how many 9's there are, because there will still be more places behind corresponding to the natural numbers after it. From what I see from below, you appear reluctant to accept an actual infinity, because given any natural number there is always one after it. But that does not exclude the possibility of an actual infinity that is not a natural number.

The bottom line is that your reductio ad absurdum relies on properties of the natural numbers, while 0.999... does not have a natural number of 9's. Thus your argument is powerless to say anything about it. All it can say is that every member of the sequence {0.9, 0.99, 0.999, ...} is less than 1, but 0.999... is not in that sequence, because the elements of that sequence can be indexed with the naturals {1, 2, 3, ...} by how many 9's there are in their expansion, and 0.999... doesn't have a natural number of 9's. You need an extra axiom indeed to say anything about it. But it is not the case that your argument proves that 0.999... < 1. Rather, starting from the Archimedean axiom you get 0.999... = 1; without it, 0.999... and its properties stand outside your system. And something like this is indeed covered by the article under #Ultrafinitism: under that view, 0.999... is considered to be meaningless because of the complete rejection of all infinities. But you will find it very hard to get much stuff done that way, which is why it does not have widespread acceptance. Double sharp (talk) 06:05, 10 December 2017 (UTC)

Dear Doublesharp. You write: “You appear reluctant to accept an actual infinity, because given any natural number there is always one after it. But that does not exclude the possibility of an actual infinity that is not a natural number. ... The bottom line is that your reductio ad absurdum relies on properties of the natural numbers, while 0.999... does not have a natural number of 9's.”

I’m sorry, but I can’t agree with you. Euclid’s theorem https://en.wikipedia.org/Euclid%27s_theorem proofs by a reductio ab absurdum that the assumption of a largest prime is illogical. As every prime is an uneven natural number and an even natural number is 1 larger than it’s preceding uneven number, the assumption of a largest natural number must illogical too. And when there is no largest natural number the multitude of natural numbers must be infinite. As infinite as the number of nines in 0.999…. Therefore, this reasoning leaves no room for a sequence of natural numbers that ceases as such and "somewhere, but can you tell me where exactly?" turns into a sequence of infinite whole? numbers.Antonboat (talk) 12:23, 11 December 2017 (UTC)

I'm not saying there is a largest natural number. I'm also not saying that the sequence of natural numbers stops somewhere. And indeed that sequence is infinite. What I am saying is that infinity is not a natural number. It is separate from them, as if you were counting "1, 2, 3, ...", and after all the natural numbers came infinity, like ω in the ordinals.

If you like, you can imagine counting "1" at t = 0.9 s, "2" at 0.99 s, and so on, and after 1 second you've counted every natural number at some point. Then after that you can deal with infinities, but you have to keep in mind that they are not natural numbers, so that properties that are true for natural numbers may not apply there. The number of 9's in 0.999... is not a natural number. So your "proof", which relies on properties of natural numbers, does not apply to 0.999..., although it does apply to 0.9, 0.99, 0.999, and so on. Does this clarify things? Double sharp (talk) 08:42, 13 December 2017 (UTC)

Dear Doublesharp. In answer to your question: “Does this clarify things?”, the following answer. You are crystal clear in your position concerning the relation naturel versus infinite numbers as you write: “What I am saying is that infinity is not a natural number.“ You are right because in my opinion infinity cannot be a number, neither natural number, nor a real number, nor an ordinal number Aleph, nor whatever other kind of number someone could assign to infinity. (But please do not take this statement very seriously. The person who writes this is a non-mathematician. And who is inclined to take a psychologist seriously when he makes mathematical statements, but not proofs them?) You write that the natural numbers are separate from the natural numbers. But I think you’ll agree with me that infinite numbers, if they could be a logical and mathematical possibility, should be > 1. And if that is true and if an infinite number is no natural number, than the infinite numbers must, according to my logic, be a follow up of the natural numbers, as in: 1, 2, 3, ..., Aleph?/ω?. N.B. If we assign an ordinal to each element of this sequence as follows: 1, 2, 3, ..., Aleph?/ω?, logically the sequence of ordinals and the sequence of naturals + infinite numbers must be identical to each other. Is this proof enough?Antonboat (talk) 14:26, 13 December 2017 (UTC)

Yes, that's pretty much how the ordinals work, extending the naturals by coming after all of them: first we have 0 as an equivalence class as (just read that as "corresponding to", if you like) the set {}, 1 as that of {0}, 2 as that of {0, 1}, and so forth. Then ω comes after all the naturals as the equivalence class of {0, 1, 2, 3, ...} = N₀, and then we have ω+1 corresponding to {0, 1, 2, 3, ... ω}, and then ω+2, ω+3, ..., ω+ω=ω2, ..., ω3, ..., ωω = ω, and then ω and all the way to ω, ω, and so forth. Then we get things like ε₀, defined as the smallest solution of the equation ω = x, and they keep going. But I don't see why you think that they can't be numbers: it really depends on what you want a number to be. Already once you get past the naturals you are generalising things, because they no longer count anything discrete. So we are taking some familiar properties of the naturals, generalising them into a bigger structure, trying to preserve some properties and gain some interesting ones, though of course we cannot keep all of them. Put like that, I don't see why ω shouldn't be a "number", if a specific kind of number.

But it's also important to note that these are not natural numbers. That's why they don't contradict the obvious argument that you can't have a largest natural number because if there was one, call it N, then N+1 is bigger. This argument is valid and true, but it doesn't rule out the possibility that you can have something that is bigger than all the natural numbers. In some systems you can even have it as the biggest object, like the extended real number line, and it works out because in that system, ∞ + 1 = ∞, so the argument doesn't hold because ∞ + 1 isn't greater than ∞. And no amount of adding together natural numbers will get you to ∞, which violates the Archimedean property (that any small number, when added to itself enough times, will surpass any larger number) that the reals have.

The reals are Archimedean: thus they don't have infinite numbers and they don't have infinitesimal numbers. (I do need to clarify that 0.999... is a real: it has an infinite number of 9's in its decimal expansion, but it isn't infinite itself: for one, it's obviously less than 2. That is all right.) So we can immediately see that 0.999... = 1 by computing 1 − 0.999... and seeing that it must be less than any positive number. (For instance, it's less than 0.0001 because 0.999999... > 0.9999 and 1 − 0.9999 = 0.0001, and you can make this argument by truncating the expansion 0.999... after any number n of digits and see that it's less than 10.) It also obviously can't be negative because 1 can't be less than 0.999... (I trust this bit is obvious). But once you say that, the difference 1 − 0.999... has to be zero, because the reals have no infinitesimals and so the only real number that's nonnegative and yet less than all the positive numbers is zero. Again, there are other systems, but the article is about 0.999... in the reals, and there it equals 1. This could be considered a corrected version of your proof at the start of this section, if you like: it takes your correct observation for decimals with finite numbers of 9's after the point, and then makes the leap to the categorically different case where there are an infinite number of 9's. Double sharp (talk) 14:49, 13 December 2017 (UTC)

Dear Double Sharp. Thank you for your many comments. It seems you comments faster than I can react. You write: ... “I don't see why ω shouldn't be a "number", if a specific kind of number. ... But it's also important to note that these are not natural numbers. “ Writing about infinite numbers you say: “But it's also important to note that these are not natural numbers. That's why they don't contradict the obvious argument that you can't have a largest natural number because if there was one, call it N, then N+1 is bigger.” We can easely see whether a number is < 1 or > 1. But how can you see which is a natural number and which is not? The multitude of natural numbers is infinite in the sense that for whatever value of the natural n we can create a number n + 1 > n.

I’m interested too in the way how to reach 1 from any 10E-n which is below any nonzero real number. You wrote (12:52, 3 December 2017 (UTC): ... “as you get further and further in the sequence (you meant: 0.999...) the distance to 1 gets below any nonzero real number you want, without reaching zero, of course.)" And you say: “the only real number that's nonnegative and yet less than all the positive numbers is zero.” I know that 0 is a digit. But how can a number be zero? By an axiom: 0 is a number? And how can you travel a distance, making steps zero units long? (NB It may be that I did not understand what you wrote, or that I did not correctly take the quotations from their context.)Antonboat (talk) 21:27, 14 December 2017 (UTC)

Dear Double sharp. Please forget the comment above because I think it is not clear enough. My intention, corresponding with you, which I appreciate very much, is:

a) to find out how mathematicians think they can succeed in achieving 0.(9) = 1, in smaller steps than the reals in the decreasing sequence: 0.1. 0.01, 0.001, ... and

b) in what way you came to accept 'infinity' as a number, and

c) in what way we can reach this number, if the naturals do not offer this, in concreto: starting from 1 , 2, 3, .... until the number 'infinity'.Antonboat (talk) 09:02, 15 December 2017 (UTC)

I really am not sure what you mean by (a). If the idea is that the numbers {0.9, 0.99, 0.999, ...} are respectively distance {0.1, 0.01, 0.001, ...} from 1, meaning that if you keep adding nines one at a time to the decimal you get closer and closer but never reach 1, that is all true. No matter how far you go in the sequence, if you stop anywhere, you don't reach 1. But "0.999..." with an infinite number of 9's means that you never stop. As for (b) and (c), before I can answer this effectively, I'd need to know: what exactly do you consider to be legitimate numbers? Evidently you don't seem to accept that 0 is a number. Do you think 1/2 is a number? −1? √2? e? i? j?

As for 0 itself: why not have it as a number? It certainly behaves consistently when you add it to the other natural numbers, just defining it as the result of 1 − 1. You can certainly do addition, subtraction, and multiplication with it, and you can even do division as long as it's the numerator and not the denominator. And in fact, the very place-value notation implies the acceptance of zero as a number that you can use as an addend: just as 583 means 5 × 10 + 8 × 10 + 3 × 10 = 500 + 80 + 3, so 503 means 5 × 10 + 0 × 10 + 3 × 10 = 500 + 0 + 3. Honestly, this sort of question feels like how the Greeks rejected 1 as a number, because a number to them had to be a multitude. Mathematicians tend not to get themselves involved in such philosophical quibbles. Rather we ask for some property we want for our original structure, and obtain it by extending that structure in some way. That is how we get from N to Z: we want a ring that we can embed the naturals in (in layman's terms, we want a structure in which subtraction is always possible). And similarly it is how we get from Z to Q: we want a field that we can embed the rationals in (in layman's terms, we want a structure in which division by anything but zero is always possible). And it helps that there are very many natural situations modelled by Z and Q that practically compel us to accept their legitimacy as structures, even as number systems, if we want to get pretty much anything done.

Now once we've accepted that, I'm having difficulty understanding what the difficulty is. Sure, you can never get to infinity by counting up with the naturals. But then again, you also can never get to 0.5 by counting up with the naturals. And why should that matter? Neither are natural numbers. The naturals don't give us everything we want, so we extended them into larger systems that do. Double sharp (talk) 16:05, 16 December 2017 (UTC)

Dear Double sharp. I knew that I was not clear enough in my comments to you. I've spent the weekend to get things more clear. This has resulted in e new section: “Why 0.999... = 1 cannot be true and infinite cannot be neither a natural, nor an infinite number.”Antonboat (talk) 21:03, 18 December 2017 (UTC)

A real larger than any real in the infinite representation of (9) 0.(9)

It is said in: https://en.wikipedia.org/0.999... “In mathematics, 0.999... (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9 after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal numbers 0.9, 0.99, 0.999, etc. “ In this comment I write instead of:${\displaystyle 0.999...}$ ${\displaystyle (0.9)}$. Following what is said above about ${\displaystyle 0.(9)}$ the infinite sequence: ${\displaystyle 0.99,0.999,0.9999,etc}$ must be represented by ${\displaystyle 0.(9)9}$. The first number in the representation of (${\displaystyle 9)9}$ is larger: ${\displaystyle 0.95}$ than the first real in the representation of ${\displaystyle (0.9)}$: ${\displaystyle 0.9}$. So in the representation of ${\displaystyle 0.(9)9}$ must be a real larger than any real in the representation of ${\displaystyle (0.9)}$.Antonboat (talk) 09:16, 11 December 2017 (UTC)Antonboat (talk) 19:10, 18 December 2017 (UTC)

I think using this notation, which avoids the explicit ellipses, seduces you to introduce a concept, not covered by the usual setting of decimals. The notation 0.999..., as said, denotes a "repeating decimal", and is not prepared to be amended to 0.999...9, because this latter notation totally loses the meaning of a "repeating decimal". It denotes a decimal with an unspecified, but finite number of 9s right to the decimal point. All this mixing up of different notations does not lead to any meaningful result. Similar holds for the somehow confused notation above of 999...9. If you saw ...999 before, this belongs to the number system of p-adic numbers, with quite different properties compared to the usual numbers, and also does not allow for 9...999.

Please, allow me to suggest that, in case you are interested in continuing beyond ${\displaystyle \omega }$ or ${\displaystyle \aleph _{0}}$, you should consult ordinal number or cardinal number, but stop tinkering with decimals. They are not apt for this task, because they are just a notational convention, no good selection for contemplating the concept of number.

And finally: sorry, but I believe you are totally wrong at this place. I think, you should consult the Help desk, or similar. Someone might close this thread, or even worse, try to delete this whole subpage, again. Purgy (talk) 10:38, 11 December 2017 (UTC)

Dear Purgy. Thank you for your suggestion. I’m a Dutch psychologist, very interested in what mathematicians say and think about infinity. And I’m more than very interested in an concrete example of the transition from a natural number to a non-natural infinite number as Aleph, as nice and simple as the way in which Euclid proofs that the assumption of a largest prime is illogical. Can you send me that? I would be very grateful to you.

As for your second comment: my note is intended as a constructive contribution to the debate about 9999 .... I expect to receive constructive and enlightening reactions like yours. And such a reaction could be that people prove to me "claire et distincte" that what I argue is pure nonsense.Antonboat (talk) 11:38, 11 December 2017 (UTC)

Dear Purgy. As an afterthought another comment. You write: “I think using this notation, (you ment ${\displaystyle 0.(9)9}$, Antonboat), .... , seduces you to introduce a concept, not covered by the usual setting of decimals." You’re right. But your interpretation of ${\displaystyle 0.(9)9}$ : ${\displaystyle 0.999...9}$ does not match with what I had in mind when I created this notation. I came up with that idea when I read in: Misplaced Pages "0.999 ..." that ${\displaystyle 0.(9)}$ is represented by the sequence: ${\displaystyle 0.9,0.99,0.999,etc}$. Analogous to that, I created the notation ${\displaystyle 0.(9)9}$ representing the sequence: ${\displaystyle 0.99,0.999,0.9999,etc}$. By means of the notation ${\displaystyle 0.(9)9}$ I wanted to indicate a repeating nine to the right of the decimal point, immediately followed by a nine, like this: ${\displaystyle 0.9...9}$. It turned out that the notation ${\displaystyle 0.(9)9}$ could be of excellent use to prove that ${\displaystyle 0.(9)}$ ≠ ${\displaystyle 1}$. The proof goes as follows:

${\displaystyle 9x0.(9)9=0.8(9)1}$. (Analogous to: ${\displaystyle 9x0.99=0.891}$).

${\displaystyle 9}$ – ${\displaystyle 0.8(9)1=0.(0)9}$. (Analogous to: ${\displaystyle 9,000-0.891=0.009}$).

${\displaystyle 0.(0)9:9=0.(0)1}$.

${\displaystyle 0.(9)9+0.(0)1=1}$.

Ergo: ${\displaystyle 0.(9)9}$, and thus ${\displaystyle 0.(9)}$, must be < ${\displaystyle 1}$.

As you write: “All this mixing up of different notations does not lead to any meaningful result.”, I do not believe that you will be very interested in my argument above. But given your fair response on my note, I thought I owed you this explanation. It seems to me very useful in a proof as above mentioned and as example of a way to multiply notations as ${\displaystyle 0.999...}$. And reading the wide variation in notation and definition in what is written about ${\displaystyle 0.999...}$ on Misplaced Pages, in my opinion, the notation ${\displaystyle 0.(9)9}$ is in this variety not out of place.Antonboat (talk) 12:50, 13 December 2017 (UTC)

Again, I must repeat: when mathematicians write 0.999... and expect the standard meaning, they don't mean the sequence {0.9, 0.99, 0.999, ...}; they mean its least upper bound.

And once again you're tacitly assuming that there's a specific finite number of digits enclosed in the parenthesis. For example, it is true that if you expand out all those parentheses into four digits, you get 9 * 0.99999 = 8.99991, then 9 − 8.99991 = 0.00009, 0.00009 / 9 = 0.00001, and then 0.99999 is obviously 0.00001 away from 1. But here's the thing: 0.999... has more than four 9's there. Well, then, put in ten nines instead, repeat those steps, end up with 0.00000000001, and 0.9999999999 is obviously 0.0000000001 (10) away from 1. But again, 0.999... has more than ten 9's there. No matter how many 9's you put in, 0.999... has more 9's than that. If you put in n 9's, you get 10 as your distance from 1. But again, 0.999... has more than n 9's, and so it's closer than 10, no matter what n you put there. Which means that the distance of 0.999... from 1 has to be 0 in the reals, as that's the only nonnegative number smaller than every negative power of ten. Does that make sense? Double sharp (talk) 14:57, 13 December 2017 (UTC)

Dear Double sharp. I have read both your comments of 13 December and my answers to some of it it may be found in the new section “Why 0.999... = 1 cannot be true and infinite cannot be neither a natural, nor an infinite number.”Antonboat (talk) 21:17, 18 December 2017 (UTC)

In the reals there are no infinities, and any infinities you see are not real numbers and don't play by their rules. In particular, we can only deal with finite negative powers of ten. You can have a 10^(-1) place, a 10^(-2) place, and so on, but there is no 10^(-ω) place there. There is only one for 10^(-n) for each natural number n, which is enough to get you an infinite number of decimal places, but not enough to continue before that. Therefore, in the reals, trying to have an infinite repeating sequence of 9's and then something else is indeed nonsense. It is indeed not nonsense in some different systems, such as the hyperreal numbers. But: (1) this article is covering the reals, which are the most generally used system, and there 0.999... = 1; and (2) the hyperreal with nines all the way even past infinity, i.e. 0.999...;...999..., still equals 1. Double sharp (talk) 09:01, 13 December 2017 (UTC)

This whole discussion highlights the problematic nature of symbol-pushing arguments such as those that, until recently, were presented as evidence of the equality in the article. Such arguments based on the manipulation of infinite decimal expansions are inherently problematic, because the mathematical notion of infinity is rather tricky. Indeed, there are many paradoxes of infinity that show that dealing with infinite objects requires extreme care (e.g., Hilbert's paradox of the Grand Hotel). In any case, one is certainly free to define the notation ${\displaystyle 0.999...}$ to refer to many different things such as an infinite expression which is manifestly not the same as the expression ${\displaystyle 1}$. A typical way to define the real number referred to by an infinite decimal expansion is the least upper bound of its finite decimal truncations. Thus ${\displaystyle 0.999...}$ is, by definition, the least upper bound of the sequence ${\displaystyle 0.9,0.99,0.999,...}$. This least upper bound is equal to one, not because of symbol-pushing arguments, but rather because of the Archimedean property of the real number system. Sławomir Biały (talk) 14:40, 13 December 2017 (UTC)

Dear Slawomir Bialy Thank you for your comment. You write: “This least upper bound is equal to one, not because of symbol-pushing arguments, but rather because of the Archimedean property of the real number system.” So it seems I am on the wrong trail with this note.Antonboat (talk) 14:56, 14 December 2017 (UTC)

I wouldn't say "wrong track". What it illustrates though is just how problematic infinite objects can be, which is part of the point of the exercise. There was a huge discussion recently about a section of the article where symbol-pushing arguments were presented as evidence of the equality, with necessary and important details left out. I and a few others were of the opinion that such arguments should not be convincing to anyone that actually needs to be convinced, because such arguments can be readily modified to prove all kinds of silliness, like ${\displaystyle ...999=-1}$. Sławomir Biały (talk) 15:17, 14 December 2017 (UTC)

@Antonboat, honestly, I want to get out of this treadmill. Double sharp seems to refer to "your" idea of ω, hidden behind that unlucky 0.(9)9, where I coined the 0.(9) as "not prepared to be amended" in the context of a "repeating decimals". The quintessence is to me that you go beyond the setting for this 0.(9). OTOH, Sławomir Biały reminds you of the correct use of "sequence ${\displaystyle 0.9,0.99,0.999,...}$", which does not cover your interpretation of "representing 0.(9)", but is just the sequence ${\displaystyle (0.(9)_{i})_{i\in \mathbb {N} }}$, which neither contains 1 nor 0.(9), but converges within the reals to 1, giving rise to the definition of the meaning for these awkward notations of real numbers 0.(9) or 0,999..., both equaling to 1.

Sorry, I cannot help you any further, but to exhort you to be more strict and coherent in using mathematical notation: There is yet absolutely no justification for abusing the standard routine for multiplying decimals of finite length for your non-decimals. "It works" is no valid argument. One last thought as good bye: 1 is the only number which has infinitely many such sequence elements in any ɛ-neighbourhood. Purgy (talk) 18:06, 13 December 2017 (UTC)

Dear Purgy. You write: “Double sharp seems to refer to "your" idea of ω, ...“, but that is not true. I used the symbol for omega in his comment, in my comment to him. You write: “There is yet absolutely no justification for abusing the standard routine for multiplying decimals of finite length for your non-decimals.” I’ll keep it in mind and won’t do it again. I now consider this note as a dead end.

You write ”Antonboat, honestly, I want to get out of this treadmill” The "mores" of Misplaced Pages assume that we Wikipedians be polite and constructive in communicating on Misplaced Pages. But we’re not obliged to respond if we do not want it or can not do that for any reason whatsoever. Thank you for your comments and: “Alle goeds “. That’s Dutch for “All the best”. — Preceding unsigned comment added by Antonboat (talk • contribs) 14:19, 14 December 2017 (UTC) Antonboat (talk) 14:52, 14 December 2017 (UTC)

So you have 0.(9)9 meaning an infinite sequence of 9's and then a 9! That reminds me of

  π goes on and on
     And e is just as cursed
     I wonder, how does π begin
     When its digits are reversed?

Dear Dmcq. Thank you for your funny respond to my idea of 0.(9)9. There is the same logic in that idea as expressed in the little poem you sent me. The notion 0.(9)9 emphasizes that we can not record the number of nines by means of the note 0.(9) because the repeating nine exceeds every representation that we can hink of. That’s why in my opinion 0.(9)9 = 0. (9). The poem states: “π goes on and on.” In the same way the nines in 0.(9) and in 0.(9)9 go on and on. “When its digits are reversed” we find for 0.(9) a natural number (9).0 that goes on and on because the smallest natural number with a nine is 9, but starting from Euclid’s Theorem, there cannot be a largest natural number consisting of only nines, nor a limit to it as the (non?)natural number Aleph, Omega or Infinite. Communicating with you and with all the Wikipedians who showed me the honor (really, I feel it that way) to respond to my notes, I noticed that you are all very much convinced of the possibility of such an infinite number. Following Euclid I do not see that possibility. To be clear: my intention is not to convert you to my beliefs. (It would be impossible I know.) I am not a missionary. My intention, corresponding with you is,

a) to find out how mathematicians think they can succeed in achieving 0.(9) = 1, in smaller steps than the reals in the decreasing sequence: 0.1. 0.01, 0.001, ... and

b) in what way you came to accept 'infinity' as a number, and

c) in what way we can reach this number, if the naturals do not offer this, in concreto: starting from 1 , 2, 3, .... until the number 'infinity'.Antonboat (talk) 08:56, 15 December 2017 (UTC)

@Antonboat: It's hard for me to follow what you're arguing, as I'm unable to see it as more than hand-waving. As Purgy said above, we are unable to have this conversation unless you are willing to propose a rigorous argument - and per WP:NOTAFORUM we shouldn't even talk about that, unless it is directly related to improving the article.--Jasper Deng (talk) 09:16, 15 December 2017 (UTC)

Dear Jasper deng. I am a native Dutchman, so I speak and write Dutch better than you do, but my English could be therefore (very) imperfect. I was not familiar with the expression: "handwaving." Searching on Misplaced Pages (thank you for your link) I found out that it means in Dutch something as: “uit je nek kletsen”, translating in English: ≈ “to chatter from your neck”. Well that’s not a friendly evaluation of my contributions to Misplaced Pages, but it’s an opinion. Thank you for that. I use this argument page tot confront Euclids theorem with what is said in Misplaced Pages “999...”. So what you see as my handwaving: in slang Dutch: “ Mijn gelul” is for me “arguing” and a serious preoccupation. Antonboat (talk) 10:48, 15 December 2017 (UTC)

Dear Jasper deng. I've started a new section: “Why 0.999... = 1 cannot be true and infinite cannot be neither a natural, nor an infinite number.” The beginning of it is as follows: "This note is a response to Jasper deng who said that my notation 0.(9)9 and what I said about it was ‘hand-waving’.Dear Jasper deng. Let me show you by simple logical reasoning that you’re very wrong in that." I invite you to react on this section.Antonboat (talk) 20:52, 18 December 2017 (UTC)

Jasper, this is the arguments page. It's specifically for people to make arguments that don't have any immediate application to the article. --Trovatore (talk) 09:25, 15 December 2017 (UTC)

Right, but WP:NOTAFORUM is a project-wide policy and at the least, it sets a limit on discussions like this one.--Jasper Deng (talk) 09:56, 15 December 2017 (UTC)

Like most of WP:NOT, NOTAFORUM is mostly about article space. (It fairly explicitly excludes the refdesks, for example.) A lot of NOT would be seriously constraining in other spaces. The limits in article-talk space come mainly from WP:TPG, which the arguments pages are specifically designed to relax. --Trovatore (talk) 10:26, 15 December 2017 (UTC)

Three questions on 0.999...

Primarily, I do not want to appear as impolite and unhelpful. It is just that I am constantly afraid of appearing such, because of my awkward use of the English language as a non-native, not even living in an English speaking country. Secondly, it is true that I want, for egoistic reasons, to reduce my involvement in this discussion, but as far as possibly without any hard feelings.

Under this preamble I want to restate my positions, which are strongly constricted to to aspects within "real numbers".

• ad a): In my perspective, the use of 0.(9) is already "hand-waving" to me (please, under all circumstances, ignore the "pejorative"-qualification, and skip to the "academic use" in the linked article!) within the reals. In my opinion, decimal representations of reals are unwieldy to explore the boundaries of theoretical properties of the reals. There is not even a bijection (proven!) between reals and decimals. Decimals gain their importance for the ubiquitous representation of real world effects, fictitiously measured within the concept of real numbers (they are rational, at best!). Any use of symbols, hinting to "infinity", like ellipsis, or (.), or ∞, ... requires a strictly defined background to allow further manipulation. All these backgrounds employ (to my knowledge) the concept of "limits", hard enough to define on the basis of abstract numbers, avoiding as much as possible their representation as decimals (free monoid, made up by the star of the alphabet of the 10 digits?). Once a specific limit-expression, accepted as definiens of the definiendum 0.(9), is found, a rigorous mathematical treatment may start. Please, note that this acceptance is via "defining a mathematical meaning for 0.(9) (= 0.999...) := blablabla" (I don't know why WP deprecates the ":="). Nobody is allowed to modify this definition within the system under discussion, without proving the equivalence, or explicitly setting up a new notion. It is simply not the case that 0.(9)9 (= 0.999...9) := blablabla9". Obviously, 0.(9)9 requires a new definition, since it violates the inaccessibility of the end of an "infinite chain" in the system of real numbers. Alluding to the poem: there is no last digit of π to start π. This is not to say that one cannot construct "numbers", which go beyond this "infinity in length" of decimals, there is, afaik, a rich hierarchy of "infinities" and unclear conjectures about their connection to the "continuum". I strongly believe, however, that there are only a few mathematicians (Wildberger?), who already deny the "infinity in length", negating the existence of reals, and the other, so I assume, do not strive for "succeeding to achieve 0.(9) = 1 in smaller steps", because, firstly, this is an equality-fullstop (according to the definition of 0.(9)!), nothing to achieve in steps, and second, there -proven!- are no smaller steps within the reals as the reals already offer (tautology?). You must change your axioms, if you want to go there (infinitesimals). BTW, they are not too handy and not widespread.
• ad b) Personally, I do not accept "infinity" as a number. As said, within the reals it is an abbreviation to me, in settings that explicitly need "limits". Sometimes the limits get so trivial that one infinity (∞), or both (±∞) are adjoined to settings of numbers, and so are treated as numbers, leading to abuse of notation like -1/0 = -∞, or 1/∞ = 0, (0, ∞). In the context of the "higher infinities" there are additional rules for calculation, and so I sometimes ask myself if these objects are rightfully coined "numbers". Possibly, the whole use of the word "infinity" in math is a revenge of philosophy for math having gotten grown up, out of this ancient guardianship: no more mathematicians to be drowned by philosophers for small irrationalities. :)
• ad c) I simply do not know, if "to be reached" is a defined property for these numbers. So, sorry, this question is "hand-waving". However, I deny "infinity" to be a number in this context. :p

Please, do not assume any hostility from my side if I stopped commenting with this. “Alle goeds!“ Purgy (talk) 16:16, 15 December 2017 (UTC)

Why 0.999... = 1 cannot be true and infinite can neither be a natural, nor an infinite number.

This note is a response to Jasper deng who said that my notation 0.(9)9 in: “A real larger than any real in the infinite representation 0.(9) “ and what I said about it was ‘hand-waving’. Dear Jasper deng. Let me show you by simple logical reasoning that you’re very wrong in that. I suspected that it would be clear to the mathematicians who read the expression 0.(9)9 in my above mentioned note that they would understand that it was a provocation, because I meant it that way. But because some reacted very shocked on it, I let it go. It was clear to me than, that I had to look for a better way to express my interpretation of 'infinity'. Let me show you that the expression 0.(9)9 in my opinion is not hand-waving. But first I’ll show you that the notation 0.999... = 1 is conflicting following the Archimedean interpretation of numbers as distances, https://en.wikipedia.org/Archimedean_property.

1) We may follow Archimedes in interpreting numbers as distances. Right?

2) Than we may interpret the equation 5 + 5 = 10 as: starting from a given point and traveling in two steps of 5 units we’ll cover a distance of 10 units. Right?

3) We may write 0.999... as: 0.(9). Right?

4) The notation 0.(9) represents the infinite sequence: 0.9, 0.99, 0.999, etc. Right?

5) The difference between 1 and 0.99 and 0.9 = 0.09, between 0.999 and 0.99 = 0.009, between 0.9999 and 0.999 = 0.0009 etc. Right?

6) These differences can be written as the decreasing sequence 0.09, 0.009, 0.0009, etc, or: as 9.10E-2, 9.10E-3, 9.10E-4, etc. (E means Exponent) Right?

7) If reals are interpreted as distances and 0.(9) as representing the sequence: 0.9, 0.99, 0.999, etc, the notation 0.(9) = 1 must mean: starting from 0.9 we can cover the distance 0.9 to 1 in ever decreasing steps: 9.10E-2, 9.10E-3, 9.10E-4, etc. Right?

8) Every possible negative power of 10 must be a negative natural number. Right?

9) Every real 10E-n must be a real > 0. (If 10E-n = 0 should be true, 1 and thus 10En too, had to be 0.) Right?

10) Because every real 9.10E-n > 0, it is impossible to merge with 1 from 0.9 in decreasing steps: 0.09, 0.009, 0.0009, etc. Right?

11) But if we cannot reach 1 by traveling from 0.9, to 0.99, to 0.999, etc, the expression 0.(9) = 1 cannot be true. Right?

N.B.1: I suggest to rewrite the notation 0.999... = 1, in agreement with the Archimedian interpretation of numbers as distances as: 0.999... ≠ 1 nearing but not reaching a limit 1.

N.B.2: Zeno of Elea showed 2300 years ago, that 1 could not be reached in decreasing units 0.5, 0.25, 0.125, etc, see: “The Dichotomy” in: https://plato.stanford.edu/entries/paradox-zeno/.

Can there be an infinite number and an actual infinity?

1) The notation 0.(9) means an infinite multitude of nines to the right of the decimal point. Right?

But can there be a sequence of nines that is ‘actual infinite’, that is: a sequence in which the infinite multitude of nines is really listed, no number nine exempted? Let us see, with the infinity of the naturals as an example.

2) Euclid’s Theorem, https://en.wikipedia.org/Euclid%27s_theorem, proofs by a a reductio ad absurdum , https://en.wikipedia.org/Reductio_ad_absurdum, that the assumption of a largest prime is absurd; illogical. Right?

3) As primes are uneven natural numbers and any even number is uneven number + 1, starting from this Theorem we may say that the assumption of a largest natural must be illogical too. Right?

4) But if there cannot be a largest natural in the infinite sequence: 1, 2, 3, etc, this sequence cannot have an upper boundary. Right?

5) In that case this infinite sequence of naturals must be understood as an ever in number growing sequence of elements without an upper boundary. Right?

6) Than ‘infinite’ in the infinite sequence of naturals can exclusively and only be understood as:‘ever growing without an upper boundary’. Right?

7) The expression: ‘infinite sequence of elements’ must than be interpreted as an: ‘ever growing sequence of elements without an upper boundary’. Right?

8) In an ‘ever growing sequence without boundary’ we cannot make the, without boundary ever growing multitude of elements, invariable by assigning a number to it. Right?

9) So an infinite: without upper boundary ever growing multitude of elements, can never loose its property: ever growing in elements without an upper boundary, by assigning a natural or hypothetical infinite number to it. Right?

10) That's why we may assign to this ever growing multitude of elements, the notion ‘infinite number’, something like: Aleph, Omega, or ∞. Right?

11) But assigning an infinite number to such a without upper bound ever growing multitude of elements like nines in: 0.(9): 0.9999999999,etc, or naturals in: 1, 2, 3, etc, cannot have any effect on the properties of this multitude. Right?

12) As this sequence is ever growing without upper boundary in elements it must than be true that, during the time you need to say: “An infinite number is represented by the symbol∞”, it could be that the number of nines on the moment you have said it, has become: ∞ + 2.10E9 and 5 minutes later ∞ + 302.10E9. Right?

13) ((Well, that’s why I think that the notation (0.9)9 does not equal nonsense. Right? (But I'll never use this notation 0.(9)9 again.))

14) That's why starting from Euclides Theorem and its consequences there cannot be an ‘actual infinite’, that is: a sequence in which the infinite multitude of nines is really listed, no number nine exempted. Right?

15) The assumption of a number 'infinite' would only be logically defensible if an 'actual' infinite could exist. Right?

16) So because, seen in the light of Euclid's Theorem and its consequences the assumption of an actual infinite is illogical, the assumption of a infinite numder must be illogical too. Right?

And for the record: and than a mathematical theory which is based on an axiom like: there is a number ’infinite’, or: actual infinity is possible, must be illogical, because such a theory is from its beginning in conflict with the consequences of Euclid’s Theorem. If I'm right in this, this position must have his impact on mathematics. For example: Cantor’s Diagonal Argument cannot be true than, because of the interpretation of ‘infinite’ as: ‘ever growing in elements without upper boundary (or shorter: without a limit) in elements’ This Argument can than be paraphrased as a a composition of the infinite, ergo ever growing, sequence of naturals: 1, 2, 3, , etc, and the infinite: ever growing sequence of reals < 1 in random order: 0.6, 0.33, 0.14, etc. From this sequences the one-to-one correspondence: , , , etc, can be prepared. Ergo: both sequences must have the same cardinality.Antonboat (talk) 20:43, 18 December 2017 (UTC)

May I point you to your habits of obstinately denying

- to keep up the difference between a whole "sequence" (established in (4)) and "single items" of this sequence, as used in other parts of your arguments

- to distinguish between the "infinity" as defined/used in the context of of this article (limits!), and the philosophical "infinite multitude" you refer to. BTW and for the records, there is no axiom that "infinity is a number" within real analysis.

- to accept that you are free to define "your infinity" to your likings, without this having any influence on the truths in other axiomatic systems. (The article to which this talk page belongs is within the realm of the standard real numbers.)

and -most of all-

- that each real number, as addressed in this article, is a full equivalence class of all sequences "converging" (involves one well defined conceptualization of "infinity"!) to the same "limit". The sequence 0.(9), according to your definition, denotes just one of these sequences, and its limit 1 is, sensibly, the representative for this very equivalence class, containing "infinitely" many other sequences having the same limit, and all of these are denoted by the same representative. (This is similar to all fractions (ax/bx) being contained within one equivalence class, which itself is the one rational number, all these fractions are equal to.)

I consider it as fully de rigeur that you deny for philosophical reasons the existence of an "infinity" as referred to within real analysis, but then you must also oppose to large parts of modern math, which the vast majority of contemporary mathematicians consider sound, useful, and interesting, even when contradicting some puristic speculations.

It is to me not useful to deny the existence of alien axiomatic systems, but within real analysis your argumentation does not hold, and the thread title is a wrongness in the first claim. Purgy (talk) 10:08, 19 December 2017 (UTC)

Dear Purgy. Thank you for your comments. You write: “May I point you to your habits of obstinately denying. “ Please do, for that’s what an “argument” page is meant for. And it could be posssible that I have “habits of obstinately denying¨, indeed. But can you speak of ‘denying’ if you come to a certain conclusion by means of : ‘logical arguing’?'

Sub a) you write you think its obstinately denying that I: “keep up the difference between a whole "sequence" (established in (4)) and "single items" of this sequence, as used in other parts of your arguments”. Why should I keep up this difference? 4) reads: 4) The notation 0.(9) represents the infinite sequence: 0.9, 0.99, 0.999, etc. What’s wrong than by stating: 5) The difference between 1 and 0.99 and 0.9 = 0.09, between 0.999 and 0.99 = 0.009, between 0.9999 and 0.999 = 0.0009 etc. Do you think that statement is not true? Please proof this. That’s what my argumentation in steps 1), 2) and more, is meant for. I tried, by what I thought to be: my logical arguing, to come to the conclusion that the expression 0.(9) = 1 and thus that 0.9 + 0.09 +0.009, etc, = 1 could not be true. Please show me where I went wrong in my arguing that 0.9 + 0.09 +0.009, etc, must be < 1.

Sub b): “to distinguish between the "infinity" as defined/used in the context of of this article (limits!), and the philosophical "infinite multitude" you refer to.” I came, by what I thought to be : my logical arguing, in the second part: Can there be an infinite number and an actual infinity? In step 1) to 6) to the conclusion that ‘infinite’, seen in the light of Euclid’s Theorem and what I thought its consequences for the properties of the naturals, to conclusion 6) : “Than ‘infinite’ in the infinite sequence of naturals can exclusively and only be understood as:‘ever growing without an upper boundary’. Right?” From Euclid’s Theorem is said on Misplaced Pages: : Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.” So I had a sound mathematical foundation to start from, I thought.

”BTW and for the records, there is no axiom that "infinity is a number" within real analysis.” (Do you know that in Dutch by writing “BTW” we mean: Belasting Toegevoegde Waarde?) In the second part of my note, after 16), I write “And for the record: and than a mathematical theory which is based on an axiom like: there is a number ’infinite’, or: actual infinity is possible, must be illogical, ...” Here is spoken of ‘a’ mathematical theory.

Sub c) “- to accept that you are free to define "your infinity" to your likings, without this having any influence on the truths in other axiomatic systems. (The article to which this talk page belongs is within the realm of the standard real numbers.)” No Purgy, I didn’t feel free. I chose to start from a sound mathematical foundation: Euclid's Theorem, see sub b) and found a different interpretation of ‘infinite’ as property of the repeating nine: 0.(9).

Sub d) “and -most of all-

- that each real number, as addressed in this article, is a full equivalence class of all sequences "converging" (involves one well defined conceptualization of "infinity"!) to the same "limit". The sequence 0.(9), according to your definition, denotes just one of these sequences, and its limit 1 is, sensibly, the representative for this very equivalence class, containing "infinitely" many other sequences having the same limit, and all of these are denoted by the same representative. (This is similar to all fractions (ax/bx) being contained within one equivalence class, which itself is the one rational number, all these fractions are equal to.)” Which definition of ‘infinite’ do you use here? If Euclid’s Theorom is right and the argumentation of the consequences of it is solid, there can be only one interpretation: an infinite repeating of the nine without upper boundary (I think).

Sub e) “I consider it as fully de rigeur that you deny for philosophical reasons the existence of an "infinity" as referred to within real analysis, but then you must also oppose to large parts of modern math, which the vast majority of contemporary mathematicians consider sound, useful, and interesting, even when contradicting some puristic speculations.”

According to: https://www.merriam-webster.com/dictionary/de%20rigueur, ‘de rigueur’ means: prescribed or required by fashion, etiquette, or custom. No Purgy, I’m a man of 75, who strives to be polite, friendly and reasonable and critical. But I’m too old to follow whatever fashion. I do not deny for fashionable reasons the ‘infinite’ as you see it. I see, following Euclid an ‘infite’ that’s neither your vision on it, nor that of the vast majority of contemporary mathematicians.” As far as I know mathematics is a discipline that is founded on well-established starting points, and I see Euclid’s theory as one of them. Please show me that ’infinite’ the way I see is a puristic speculation.

Sub f) “It is to me not useful to deny the existence of alien axiomatic systems, but within real analysis your argumentation does not hold, and the thread title is a wrongness in the first claim.” Please show me, by logically reasoning the number of the argument where, according to you, I was wrong. — Preceding unsigned comment added by Antonboat (talk • contribs) 12:34, 20 December 2017 (UTC)

@Antonboat: Do you know the definition of a "sequence"? I have debunked the rest of what you said below.--Jasper Deng (talk) 12:45, 20 December 2017 (UTC)

According to Misplaced Pages "A sequence can be thought of as a list of elements with a particular order". Why did you ask this? Regarding your comment:"I have debunked the rest of what you said below." What rest have I said below?Antonboat (talk) 13:07, 20 December 2017 (UTC)

@Antonboat: Hence why I think you lack the necessary prerequisites for having a mathematical discussion. You need to know the rigorous definition of a sequence as a function from the natural numbers to some other (nonempty) set. A net is a generalization to where the domain need not be the natural numbers. And as for my "debunked" phrase, I meant that I debunked your initial comment in this section with the comment I made below.--Jasper Deng (talk) 13:10, 20 December 2017 (UTC)

@Antonboat: Sorry, TL; DR: what you wrote above is not even wrong, for the reasons stated by Purgy.--Jasper Deng (talk) 11:36, 20 December 2017 (UTC)

And if I may add to what Purgy said:

1. Before you can have a conversation about the real numbers at this level of abstraction, you have to work with a construction of the real numbers, many of which use an equivalence relation of some sort.
2. Point 7 in the first list of arguments is a bona fide example of hand-waving. You need a meaningful and rigorous definition of "covering a distance" before you can attempt to make an argument with it.
3. Point 8 above is blatantly incorrect. Natural numbers can only be integers and you are at the minimum talking about rational numbers.
4. Point 10 is also hand-waving, because you did not define "merge with".
5. Hence point 11 is meaningless.
6. In your second list, points 2 and 3 do not need Euclid's theorem to prove. The natural numbers being an infinite set follows from the very definition of finiteness: a set is finite if and only if it can be put in bijection with a bounded subset of the natural numbers.
7. Point 6 in your second list is hand-waving. A rigorous definition of "infinite" when referring to the cardinality of a set is that there exists no surjective function from a bounded subset of the natural numbers to that set.
8. In light of the previous point of mine, your point 10 is a misinterpretation of the accepted meaning of ordinals like aleph-naught.
9. For point 11, as has been emphasized repeatedly (even by yourself), infinity is not a natural number. It is incorrect to assert that adding an "infinity" to the natural numbers will preserve the properties of the natural numbers defined as-is - notably, as you yourself have pointed out, such a new number system could not enjoy the Archimedean property that you emphasize so much.
10. For point 12, you are once again hand-waving. The notion of any number plus infinity is not meaningful unless you choose a definition of how you are extending the addition operation to operate on "infinity".
11. For your remaining points, whether "infinity" is considered to be a number per se is a matter of convention. However, it is a well-defined concept.

I don't mean to come off as rude, but in order to talk with mathematicians, you need to, in a sense, speak the language of mathematicians. Learn to make everything you say well defined. This is all in addition to what Purgy said above, and not in conflict with or in place of it.--Jasper Deng (talk) 12:10, 20 December 2017 (UTC)

Dear Jasper deng. I have no time now to respond to your comments. For now, you write:"I don't mean to come off as rude, but in order to talk with mathematicians, you need to, in a sense, speak the language of mathematicians." You may be rude to me I'm a psychologist and used to that, I earned my money with it. But what's your intention with your rudeness? Making me angry, afraid, unsure? Are you unable to respond politely? Than I have to accept that. But why not try to show me with logical arguments instead of rude words that I am wrong. Please try it.Antonboat (talk) 13:34, 20 December 2017 (UTC)

@Antonboat: I just gave you the exact logical flaws in your argument. And I would love to "fix" your argument but at this time, it is insufficiently rigorous for me to follow your path. The reason for the dissection is to hopefully give you a sense of the standards expected for a discussion of this sort, because we always love seeing new ideas - but only when those ideas are stated in an adequate framework, which yours unfortunately is not. Part of it, I guess, is that I'm irked when I see a highly flawed argument passed off as "logical reasoning". If you are unsure of the footing of your argument, then by all means ask the necessary questions and do the necessary research. Wrongly claiming that I'm "very wrong" is going to be met by showing the exact opposite; pot meet kettle.--Jasper Deng (talk) 13:38, 20 December 2017 (UTC)

Dear Jasper deng. These are my comments on the comments you made in Jasper Deng (talk) 12:10, 20 December 2017 (UTC). Thank you for your evaluation that my note you comment on is ‘not even wrong’.

Sub 1: You write: “ Before you can have a conversation about the real numbers at this level of abstraction, you have to work with a construction of the real numbers, many of which use an equivalence relation of some sort.” Why must I do that in my note while it isn’ t done by the author of Misplaced Pages “0.999...”., from which article I copied the row/series/sequence 0.9, 0.99, 0.999, etc, as a representation of 0.(9), and by the wikipedians which reacted on my notes? For example Double sharp wrote to me: “Sequence 0.999... can be defined as the limit of that sequence (0.9, 0.99, 0.999, ...), i.e. a number separate from the sequence which these terms get as close as you want to (but might not reach). Double sharp (talk) 12:52, 3 December 2017 (UTC). And above all, why do I have to do that when the numbers I used in the first part of my note are repesented by distances? Because I’m no mathematician? Or are you nitpicking here?

Sub 2: You write: “Point 7 in the first list of arguments is a bona fide example of hand-waving. You need a meaningful and rigorous definition of "covering a distance" before you can attempt to make an argument with it.” Is “travel” better instead of “cover”? But Is such a meaningful and rigorous definition of "covering a distance" needed here. Would this lead to different result than I found in 7)?

Sub 3: You write: “Point 8 above is blatantly incorrect. Natural numbers can only be integers and you are at the minimum talking about rational numbers.” Please read this line once again but now good! I’m speaking there of the negative powers of 10 in line 7) and these are all negative natural numbers.

Sub 4: You write: “Point 10 is also hand-waving, because you did not define "merge with".“ You’re right, I could not find the right word for the Dutch expression: ‘in elkaar opgaan’. Is ‘unite’ a better word for a situation in which the distance between a real seen as a distance and 1 = 0? And, is the conclusion in 10) wrong?

Sub 5: You write: “Hence point 11 is meaningless.” But if I write instead of: “But if we cannot reach 1 by traveling from 0.9, to 0.99, to 0.999, etc, the expression 0.(9) = 1 cannot be true.”: “But as there cannot be a real in the row 0.9, to 0.99, to 0.999, etc, that has a distance = 0 to 1 we can never unite with 1 traveling from 0.9, to 0.99, to 0.999, etc.”

Sub 6: You write: “In your second list, points 2 and 3 do not need Euclid's theorem to prove. The natural numbers being an infinite set follows from the very definition of finiteness: a set is finite if and only if it can be put in bijection with a bounded subset of the natural numbers.“ So 2) and 3) are true?

Sub 7: You write: “Point 6 in your second list is hand-waving. A rigorous definition of "infinite" when referring to the cardinality of a set is that there exists no surjective function from a bounded subset of the natural numbers to that set.” And than this unbounded set of natural numbers must be seen as: ever growing without an upper bound, isn’t it?

Sub 8: You write: “In light of the previous point of mine, your point 10 is a misinterpretation of the accepted meaning of ordinals like aleph-naught.” Oh no, I just named it and didn’t even try to define it. But if I should try it I would start this, by the definition of an ordinal which is given in: https://en.wikipedia.org/Ordinal_number in which is written: “Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on.” And it follows from that definition that all the ordinals assigned to an infinite multitude of naturals wihout an upper boundary must also be naturals.”

Sub 9: You write: “For point 11, as has been emphasized repeatedly (even by yourself), infinity is not a natural number. It is incorrect to assert that adding an "infinity" to the natural numbers will preserve the properties of the natural numbers defined as-is - notably, as you yourself have pointed out, such a new number system could not enjoy the Archimedean property that you emphasize so much.“ I did not argue that there. I’m saying there: if you wish, you may assign an infinite number tot this ever without upper bound growing number. But in such way you can not change the property of such an ever growing multitude of nines or naturals. N.B.: I never emphasize in this note the Archimedean property. Following Archimedes I interpret numbers as distances, see 1) in the first part of my notation.

Sub 10: You write: “For point 12, you are once again hand-waving. The notion of any number plus infinity is not meaningful unless you choose a definition of how you are extending the addition operation to operate on "infinity". Oh no my dear Jasper deng. Read 12) once again but now good! What I’m saying there is: you may assign any number or symbol to an infinite without upper boundary growing multitude, but that must be meaningless, because after that, this multitude is stil growing, because of Euclid’s Theorem and its consequences, as seen above, see second part of this note 2) to 7).

Sub 11: You write: “For your remaining points, whether "infinity" is considered to be a number per se is a matter of convention. However, it is a well-defined concept. “ Why don’t you comment on 13) to 16) . Do you agree with them? If not, I invite you here to comment on this items.

Dear Jasper deng. What I meant with this note is a concept to create a logical framework to proof the idea a) that 0.999... = 1 could not be right if reals would be seen, following Archimedes as distances, and b) that starting from Euclides Theorem and his consequences there could not be an actual infinite and no infinite numbers.

Evaluation of your comments. You have shown me that when I used some expressions, they were not always in the right mathematical terms. Thank you for that, but I suspected that already, see above. You write: “ I don't mean to come off as rude, but in order to talk with mathematicians, you need to, in a sense, speak the language of mathematicians. Learn to make everything you say well defined. “ What you say here is, I think, not fair arguing. What would you say if I told you that your arguments have no value for me because they were not written in perfect Dutch? Would I be rude to you if I should say that you’re often “nitpicking” in your comments by: fussy or pedantic fault-finding? In my opinion you do this in a effort to slalom the really important ideas in this note. In my opinion you barked loudly, but you didn’t bite me. Please bite me.

- Show me that the infinite multitude of naturals has some upper boundary and may be seen as a set which is actual infinite.

- Bite me and show me that aleph naught can not be a natural number.

- Bite me and show me that we may not interpret the infinite of the natural numbers as: an ever growing multitude without an upper boundary.

- Please bite me deep, and show me that Cantor’s Diagonal Argument can be true, if ‘infinite’ is interpreted as: ‘ever growing in elements without upper boundary (or shorter: without a limit) in elements’ .

-Bite me still deeper by showing me that my parafrase of this Argument that it must be seen as a a composition of the infinite, ergo ever growing, sequence of naturals: 1, 2, 3, , etc, and the infinite: ever growing sequence of reals < 1 in random order: 0.6, 0.33, 0.14, etc., and its conclusions: “From this sequences the one-to-one correspondence: , , , etc, can be prepared. Ergo: both sequences must have the same cardinality”, must be wrong. From this sequences the one-to-one correspondence: , , , etc, can be prepared. Ergo: both sequences must have the same cardinality.", is not true.

Shakespeare wrote: “What's in a name? That which we call a rose by any other name would smell as sweet.” , https://www.brainyquote.com/quotes/william_shakespeare_125207. Does the sequence 1, 2, 3, etc change in properties because I call it a series or a row? Don’t you understand what I mean if I write: “7) If reals are interpreted as distances and 0.(9) as representing the sequence: 0.9, 0.99, 0.999, etc, the notation 0.(9) = 1 must mean: starting from 0.9 we can cover the distance 0.9 to 1 in ever decreasing steps: 9.10E-2, 9.10E-3, 9.10E-4, etc.” If you think the word “cover” is not the right word here, why don’t you give me a better, without changing the meaning of this sentence.Antonboat (talk) 09:39, 21 December 2017 (UTC)

@Antonboat: In one word: rigor. If you're going to resist and deny the need to be rigorous, you'll get nowhere, and we will not entertain having this conversation. The simple fact is that what you are saying is too vague for me to infer anything meaningful. Misplaced Pages's articles are not meant to always give a fully rigorous definition in the first sentence, but (ideally) such a definition is always found later in the article, and because you are trying to make an argument here, you are required to use mathematical language. Because the simple fact is that I could infer nothing meaningful from what you are saying. Although the notion of "distance" is formalized by the notion of a metric space, in the physical world one probably cannot meaningfully talk about "distance" shorter than the Planck length, and at this time, the way you use the word "distance" is too hand-wavy for me to infer anything meaningful from it. Whether real numbers should be thought of as representing "distance" is a matter of convention, but what is not acceptable is using it in such a vague and imprecise manner. This is mathematics, not Dutch, so the nitpicking here is going to be mathematical in nature. And the nitpicking is necessary because right now, you are saying absolutely nothing meaningful. This is completely fair because the burden is on you, as the arguer, not me, to make a coherent argument.

I also think you completely miss the notion of convergence of a sequence, along with its relation to "infinity" and what Purgy has said below about a sequence and its limit, and that before continuing further you must familiarize yourself with the precise definition of convergence. What Double sharp said there was meant to provide you with intuition, but as a definition, is hand-waving and insufficient for an argument of this sort. While it is true that no term of the sequence is exactly 1, its limit is 1 and we define the notation 0.(9) to denote the limit of the sequence, not any particular term. That argument has been presented numerous times on this very talk page and soundly debunked every single time: you are conflating members of the sequence with the sequence's limit. And no, Cantor's diagonal argument completely refutes your above argument attempting to show that the reals are countable: no matter how you try to enumerate the reals I can give you another that wasn't in your sequence. I also think you miss the point that there exists no sequence that enumerates all the real numbers - did you actually read the article on the argument? Both the reals and natural numbers are sets of infinite cardinality, but the former is far bigger than the latter, and you are making a circular argument by assuming a priori that the reals can be represented as a sequence in the first place. It is also not up to me to guess what you are trying to say by getting "closer" - your sentence had no meaning in the first place, so there is nothing I can do to "correct" it.

I accept wholeheartedly that (as precisely stated), the cardinality of the set of natural numbers is infinite, namely aleph-naught. But you cannot operate on infinity like a number. Thus your "∞ + 2.10E9" is a meaningless expression. "Ever growing without upper bound" is hand-wavy and you should wean yourself off using that phrase, since while it is acceptable as a way of gaining intuition, it is not acceptable as a definition of "infinite" - I already gave you a rigorous one and you should try to embrace it. The existence of the concept of "infinity" does not, as explained to you repeatedly, contradict Euclid's theorem. That theorem says nothing about extending the naturals to include infinite members. Your point 9 above is in direct contradiction to what you have said here. And ultimately, if 0.(9) isn't 1, then what is it?

I could go on and on about all the problems with your argument, but I will stop here. It's hard to tell if you have been trying to embrace the requirements of talking about this subject. You have been told numerous times that absent sufficient rigor, we are left scratching our heads about what you are trying to say. I recall you, as a psychologist, wanted to see our thought process here. We could give you an informal description of it, but I sense you want the exact steps. However, for the latter, we are going to talk in the language we speak, and it's up to you, not us, to ensure you understand it.--Jasper Deng (talk) 17:46, 21 December 2017 (UTC)

Dear Jasper deng. Thank you for your comments.

You write: “While it is true that no term of the sequence is exactly 1, its limit is 1 and we define the notation 0.(9) to denote the limit of the sequence, not any particular term.” That’s what I meant with my suggestion: N.B.1: I suggest to rewrite the notation 0.999... = 1, in agreement with the Archimedian interpretation of numbers as distances as: 0.999... ≠ 1 nearing but not reaching a limit 1. “ That would , in my opinion, prevent a lot of comments, to the Argument page in the future. (I’ve deleted the words : “in agreement with the Archimedian interpretation of numbers as distances“ here because they are not relevant for the definition of 0.999....)

You write: “That argument has been presented numerous times on this very talk page and soundly debunked every single time: you are conflating members of the sequence with the sequence's limit. “ No I did not and I do not, see my definition above..

You write: “And no, Cantor's diagonal argument completely refutes your above argument attempting to show that the reals are countable: no matter how you try to enumerate the reals I can give you another that wasn't in your sequence.” Yes and I can give you a natural number to serve as an ordinal for every real that you create in Cantor's diagonal argument, starting from Euclid’s Theorem and its consequences.

You write: “I also think you miss the point that there exists no sequence that enumerates all the real numbers - did you actually read the article on the argument? “ I did read a lot of articles even in German and even the interpretation of Douglas Hofstadter in: “Gödel Escher Bach”. I never could have missed this point because I see the infinite series of reals as ever growing without an upper boundary. Following Euclid’s Theorem and its consequences, I argued in this note: “Cantor’s Diagonal Argument cannot be true than, because of the interpretation of ‘infinite’ as: ‘ever growing in elements without upper boundary (or shorter: without a limit) in elements’ This Argument can than be paraphrased as a a composition of the infinite, ergo ever growing, sequence, read here: “series” of naturals: 1, 2, 3, , etc, and the infinite: ever growing sequence, read here instead: “series” of reals < 1 in random order: 0.6, 0.33, 0.14, etc. From this sequences /series the one-to-one correspondence: , , , etc, can be prepared. Ergo: both sequences/series must have the same cardinality.”

You write: “Both the reals and natural numbers are sets of infinite cardinality, but the former is far bigger than the latter, and you are making a circular argument by assuming a priori that the reals can be represented as a sequence in the first place.” I've never said that the reals could be represented by a sequence/series of all!! the reals. From Euclid’s Theorem follows that there can’t be no largest natural number, thus although the number of natural numbers and of the reals are infinite: ever in mutitude growing without an upper boundary, the logical consequence of it must be that there cannot be an infinite series of all the naturals none exempted, and nor of the reals. You write: “you are making a circular argument by assuming a priori that the reals can be represented as a sequence in the first place.” No, I didn’t , but Cantor, which I hold in high esteem because of its pioneering research concerning infinity, presents (not: 'represents’ Jasper!) in his Argument the reals as a series in random order.

As an aside: You write: “ ... in the physical world one probably cannot meaningfully talk about "distance" shorter than the Planck length, ... ” In: https://en.wikipedia.org/Planck_length, is said “Much like the rest of the Planck units, there is currently no proven physical significance of the Planck length.” If this unit is real and > 0 it can be divided. If it’s pure hypothetical it can be illusive, and such units you can only divide by illusive dividing.

An extremely short summary of all my contributions on this Argument page would be: Antonboat is arguing that there are as many possible nines and thus naturals, as there are reals. I therefore suggest that our communication will be limited to the question of whether there are just as many nines, so naturals, as reals.

I must say I appreciate your way of communicating very much, but not to much hand-waving please, it might hurt your wrist.

Merry Christmas to you to and to all the Wikipedians that commented on my contributions.Antonboat (talk) 15:51, 22 December 2017 (UTC)

@Antonboat: You are free to disagree with the definition of 0.999... as the limit of that sequence, but then in order to even compare it to a given number you have to still assign a meaningful value to it. Hence ${\displaystyle 0.999...\neq 1}$ is a meaningless statement under what you are trying to suggest. Regarding distances, the Pauli exclusion principle still sets limits on how much distance we can measure. And you are yet again flatly incorrect about Cantor's diagonal argument. I repeat, the reals cannot be enumerated by a single sequence, and assuming otherwise is plainly circular reasoning - you are basically assuming that all infinite cardinalities are equal, which is patently false. In mathematics, you have to clearly, succinctly, and precisely state every assumption you make, no exceptions. In fact you can't enumerate the reals with countably many sequences either. You cannot claim that you can provide a natural number for every real number; in the particular case of the number Cantor constructs as a counterexample, the presumption is that you have already mapped every natural number to a real number, so you are no longer free to then assign another natural number to that counterexample. If you do assign a particular natural number then you have to give up the assignment you had earlier made for it; either way you failed to cover all the real numbers. Also, strictly speaking, we are not talking about infinite series here with your particular example here, we are talking about infinite sequences. If you want to use the reals as the indexing set rather than the naturals, then we can't talk about sequences anymore; we have to talk about nets, and I suggest you avoid that given your current state of confusion.--Jasper Deng (talk) 20:04, 22 December 2017 (UTC)

Dear Jasper Deng. You write: “You cannot claim that you can provide a natural number for every real number; in the particular case of the number Cantor constructs as a counterexample, the presumption is that you have already mapped every natural number to a real number, so you are no longer free to then assign another natural number to that counterexample.”

But yes I claim I can. Give me any real 10E-n as a starting point and I’ll show you how you can do that.

Indeed Cantor presumed, but did not prove!, in his diagonal argument, that he had already mapped every natural number to a real number. Since when is an assumption! of something in science accepted as a “proof”?Antonboat (talk) 12:08, 2 January 2018 (UTC)

No you can't. And Cantor was probably not talking about what you think he was, and if he was, then he was talking about the nonexistence of such. When I say "presumption", I am talking about the hypotheses Cantor starts with before beginning his proof by contradiction (which always starts with an assumption, namely the negation of what you are trying to prove).--Jasper Deng (talk) 20:08, 2 January 2018 (UTC)

The gold standard for a proof nowadays is that not only has it been peer reviewed but that one has passed it though Automated proof checking. This does not stand for hand waving type arguments or fuzzy thinking. This starts off with a set of axioms and rules for deduction and gets rigorous proofs. Different axioms and rules can led to different things but there is a pretty much agreed part for everyday work rather than testing what can happen. The standard real number system where 0.999... = 1 and Ordinal numbers which include various infinities are part of that, and it is extremely unlikely that someone is going to come along and find some contradiction in them. So what I see you as basically saying is you don't like people bandying around infinity as a real thing. Well that's tough. Mathematicians find the concept works fine and is useful. As David Hilbert said "No one shall drive us from the paradise which Cantor has created for us." Dmcq (talk) 17:05, 20 December 2017 (UTC)

Dear Dmcq. You write: “The standard real number system where 0.999... = 1 and Ordinal numbers which include various infinities are part of that, and it is extremely unlikely that someone is going to come along and find some contradiction in them.” I’m sure that what you say here is true. But the theories that contain different infinities must be based on the axiom: there is an actual infinity. Seen in the light of an mathematically accepted proof, like Euclid's Theorem and the consequences i've been described, see for example in: Can there be an infinite number and an actual infinity?, the points 1) to 7) the natural numbers must be an ever, without an upper boundary, growing number. According to mathematicians here in Holland, mathematics cannot manage such a concept, but very well the axiom: There is an actual infinity. But this axiom is and stays in conflict with the consequences of Euclid's Theorem, whether you agree with it or not.Antonboat (talk) 09:16, 22 December 2017 (UTC)

@Antonboat: Nope. As Purgy has said before, "infinity exists" is not an axiom that defines the real numbers and without a definition of "infinity", is not a well-defined statement. The closest precise concept is the axiom of infinity (yes, please read that article in full: it doesn't say what you are trying to say). And yet again, you are plainly incorrect in whatever you might mean by "this axiom is and stays in conflict with the consequences of Euclid's Theorem". The theorem, after all, already uses the notion of an infinite set, and in particular is a statement that the prime numbers form such a set. Your statement "the natural numbers must be an ever, without an upper boundary, growing number." is absolutely meaningless, and appears to conflate the set of natural numbers with the members of that set. This is why we can't infer anything meaningful from your above comment(s).--Jasper Deng (talk) 09:22, 22 December 2017 (UTC)

No jasper deng, if you accept Euclid’s Theorem as proof that the assumption of a largest prime is absurd: illogical, you have to accept the same for the naturals. In that case a set of all!! naturals is impossible. Again: Merry Christmas..Antonboat (talk) 16:44, 22 December 2017 (UTC)

@Antonboat: You are for the umpteenth time flatly wrong about this, unless you can prove that the axiom of infinity is inconsistent with the other axioms of ZFC set theory.--Jasper Deng (talk) 20:04, 22 December 2017 (UTC)

Dear Jasper Deng You write: “You are for the umpteenth time flatly wrong about this, unless you can prove that the axiom of infinity is inconsistent with the other axioms of ZFC set theory.” I’ m sure that this axiom is not in conflict with the other axioms. But can you prove me that the axiom of infinity is true?Antonboat (talk) 12:22, 2 January 2018 (UTC)

An axiom cannot be proven or disproven. But you can't have it both ways. The natural numbers are by definition the object (a set in particular) whose existence is asserted by the axiom of infinity.--Jasper Deng (talk) 20:00, 2 January 2018 (UTC)

As a newcomer to the discussion, I think I may help you with your intuitions on infinity, as I struggled with the concept for some time as well. A similar case for me was the concept of the imaginary number i, which is defined as the square root of -1, a negative number. While it's true that it's impossible to calculate the imaginary unit from the basic operations introduced before for real numbers (addition and multiplication), that doesn't prevent mathematicians from treating it as a mathematical object and study their properties, which happen to be quite useful in practice. As long as no contradictions appear by declaring that such object exist, you're creating a new area of mathematics that is not incompatible with the previous principles, but that is an extension of them.

Another example is with the assertion that "the sum of the angles of a triangle is 180º"; if you study what happens if you declare that this assertion is false, you get the concept of non-euclidean geometry. Infinity should be handled using the same intuitions: its existence does not follow from the basic operations of finite and finitely enumerable numbers, but these operations don't prevent its existence either; therefore, the mathematics that appear from declaring that such object exists are useful and worthy of study.

Under this light, your reasoning above ("if there's no largest natural, then a set containing all naturals is impossible") is a non-sequitur: the consequent doesn't follow from the premise; the only consequence from that premise is that such set is not finite, not that it doesn't exist.

In fact you can define such a set in a rigurous way. If I remember my basics from early programming theory courses, a set can be defined by intension by defining a test function - a property that all its members possess, and only its members. In the case of naturals, that property is "it can be calculated by repeatedly applying the operation "successor" to 0 a finite number of times". No mater how high is the natural number you give me, I theoretically could confirm that it belongs to the naturals by applying my function. Then I can define the set of all Naturals as "the set of all objects that satisfy my above test function (and only those)". In this case, the equivalent of Euclid's Theorem for natural numbers shows is that this set can not possibly be finite (if you say that any particular natural number n is the last one in the set, I can create a new natural number larger than that: n' = successor(n), which is a natural because it passess the test function). Diego (talk) 20:16, 22 December 2017 (UTC)

Dear Diego.You write: “Under this light, your reasoning above ("if there's no largest natural, then a set containing all naturals is impossible") is a non-sequitur: the consequent doesn't follow from the premise; the only consequence from that premise is that such set is not finite, not that it doesn't exist.” In my opinion the consequence of Euclid’s Theorem is that as there can’t be a largest natural, the infinite set of naturals must be interpreted as a finite set of naturals to which boundlessly are added new, unique naturals.Antonboat (talk) 11:51, 2 January 2018 (UTC)

You said it: that's an opinion, not a logical inference. Thus, the consequence does not result from a formal proof from the steps before it. In mathematics, every step in a proof must follow from the axioms, forming an unbroken chain of simple and solid reasoning steps, using a well-defined set of inference rules; there's no place to opinion (other than the act of choosing the initial set of axioms). Diego (talk) 22:55, 2 January 2018 (UTC)

@Antonboat: I should also add that I mean it when I ask you to drop your hand-wavy "definition" of "infinity" as "ever growing without an upper boundary". The set of real numbers between 0 and 1 exclusive is a bounded subset of the real numbers and yet is of infinite cardinality. It seems to be blinding you to what mathematicians mean by "infinite": a given set is said to be of infinite cardinality precisely if there exists no surjective function from any bounded subset of the natural numbers to that set.--Jasper Deng (talk) 09:49, 23 December 2017 (UTC)

Dear Jasper Deng. You write: “The set of real numbers between 0 and 1 exclusive is a bounded subset of the real numbers and yet is of infinite cardinality. It seems to be blinding you to what mathematicians mean by "infinite": a given set is said to be of infinite cardinality precisely if there exists no surjective function from any bounded subset of the natural numbers to that set.” A bounded subset of the real numbers that could be of infinite cardinality, isn’t that a contradictio in terminis, https://en.wikipedia.org/Contradictio_in_terminis?Antonboat (talk) 12:40, 2 January 2018 (UTC)

No, not at all. Read the definition of a bounded set. The word "size" in the lead of that article means something different from the number of elements in it.--Jasper Deng (talk) 20:00, 2 January 2018 (UTC)

I also seem to have used too tedious a formulation in writing that you deny to strictly keep up meanings of notions, even of those you defined yourself (sequence vs. one of its items). The simple reason is that by this you break any conclusive argument you might have had in your intentions. Either you use single elements of this sequence, which are all <1, or you use the whole sequence, which itself has no value, but stands for a value that fits to a certain setting by axioms, which are missing from your statements. Therefore, setting aside all the rubbish from (5) to (10), your claim (11) is rendered meaningless, since no value for 0.(9) has been defined along your way. Math does not allow to simply state the result of a sum of "infinitely many" summands having some property. For e.g., summing 1/n grows beyond all bounds, but summing the reciprocals of arbitrarily small powers (>1) of n does converge, in the sense relevant within reals, and to which you should make yourself acquainted with.

In no way you should assume yourself not to be refuted, just because it is too tedious, as correctly said already, to reply to "insufficiently rigorous" claims.

BTW, you cannot brag with your age considering mine. What does this suggest to you? Purgy (talk) 09:20, 21 December 2017 (UTC)

Dear Purgy. I begin with an answer to your last question. I get the idea that I must say: “thou” instead of “you”, but thou implies: “Thou“. Therefore I leave it so and say “you.” I accept that I’m growing old, but I don’t brag with it. Why should I ? You write: “In no way you should assume yourself not to be refuted, just because it is too tedious, as correctly said already, to reply to "insufficiently rigorous" claims”. When I was young, some years ago, I wanted to reach the unreachable like Don Guijote de la Mancha. But now I’m to old and maybe too unwise for that. Besides I have to sing and play with our grandson who is somewhat more than two years old. But I think it is a pity that I can not introduce an, according to me sound principle as Euclid's Theorem and its consequences, in mathematics because I miss the necessary mathematical knowledge. Dear Purgy what advise can you give me in this matter? — Preceding unsigned comment added by Antonboat (talk • contribs) 10:29, 21 December 2017 (UTC)

For starters I'd start by giving up thinking about mathematics as being a part of physics, that distance in mathematics has a direct link to distance in the real world. Penrose for instance talks about three worlds - the actual world, the world of physical theories, and the world of mathematics, see for a description. And there where they say (the student, not Penrose) "I am not sure if there may be some mathematical truths that are simply beyond human reason, though", we have proved there are parts of mathematics which are inaccessible to us. Dmcq (talk) 11:04, 21 December 2017 (UTC)

Dear Dmcq. In my note I meant: ‘travelling distances’, where fysical distances are expressed in mathematical numbers. You write: “... we have proved there are parts of mathematics which are inaccessible to us.” In that case I think: Be carefull!, but than I become curious too. There are lots of things which are inaccessable to us, for example: to know what is the essence of electricity. But the electrical laws can be excellently expressed in mathematical terms. So if there are parts in mathematics inaccessable to any mathematician, I think there can be reason to distrust, because I think there is a big chance then that these parts are as illusive as the speaking wolf in Little Red Riding Hood.Antonboat (talk) 10:03, 22 December 2017 (UTC)

A good example of what I was trying to say. You are mixing up your conception of the physical world with actual physics and you are mixing that up again with mathematics. In current physics real numbers are sill used to model distances but we're not a all certain it means much at very small distances because of limitations from quantum mechanics. Our physical conception has changed because of experimental data. The mathematical idea of the standard real number system is unaffected by this and does not intrinsically have anything to do with physical distance, it just was based on the original naive conception of distance in the real world but actually is now based on axioms. Dmcq (talk) 14:09, 22 December 2017 (UTC)

Dear Dmcq. In an effort to prevent further discussion on ’distances’ the following. Jasper deng, Jasper Deng (talk) 17:46, 21 December 2017 (UTC), wrote to me: “While it is true that no term of the sequence is exactly 1, its limit is 1 and we define the notation 0.(9) to denote the limit of the sequence, not any particular term.” I answered to this: That’s what I meant with my suggestion: N.B.1: I suggest to rewrite the notation 0.999... = 1, as in agreement with the Archimedian interpretation of numbers as distances: 0.999... ≠ 1 nearing but not reaching a limit 1. “ That would, in my opinion, prevent a lot of comments to the Argument page in the future. (I’ve deleted the words : “in agreement with the Archimedian interpretation of numbers as distances“ here because they are not relevant for the definition of 0.999....)

Jasper deng wrote me about distances: ... in the physical world one probably cannot meaningfully talk about "distance" shorter than the Planck length, ... ” My response to it was: In: https://en.wikipedia.org/Planck_length, is said “Much like the rest of the Planck units, there is currently no proven physical significance of the Planck length.” If this unit is real and > 0 it can be divided. If it’s pure hypothetical it can be illusive, and such units you can only divide by illusive dividing.

I wish you a Merry Christmas.Antonboat (talk) 16:31, 22 December 2017 (UTC)

I was trying to answer " I miss the necessary mathematical knowledge. Dear Purgy what advise can you give me in this matter?" Dmcq (talk) 18:28, 22 December 2017 (UTC)

@Antonboat: You can get a real number by dividing the Planck length by something greater than 1, but the hypothesis is that there can exist no two objects whose distance is that, because two objects that close don't have a well-defined distance.--Jasper Deng (talk) 20:04, 22 December 2017 (UTC)

Yes, merry Christmas to all, who believe in Cantor's diagonal, but also to intuitionists, constructivists, (ultra-)finitists, and also to continuous classical and discrete quantum- and string-, and loop-gravity-physicists, to all Hausdorffians and non-compacts, in wild spaces, to those in Minkowski spaces or Calabi-Yau-manifolds.

I hope math can survive that when 0.(9) < 1, then also

${\displaystyle e>\displaystyle \sum _{n=0}^{\infty }{\dfrac {1}{n!}}\quad }$ and ${\displaystyle \quad \pi ^{2}>6\cdot \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\quad }$:)

I'll light the 4-th candle on my Advent wreath, and, even when ungendered: "Peace on earth, and good will to men". Purgy (talk) 17:50, 22 December 2017 (UTC)

Dear Purgy. Thank you for your good wishes. There can be miracles when you believe. (The Prince of Egypt.)Antonboat (talk) 12:51, 2 January 2018 (UTC)

A Christmas cracker joke from Marcus du Sautoy

Teacher: Today we're learning about large numbers. What does anyone think is the largest number?

A pupil: Seventy three million five thousand and twelve.

Teacher: How about Seventy three million five thousand and thirteen?

Pupil: Well, at least I was close!

:) Dmcq (talk) 23:52, 24 December 2017 (UTC)

Dear Dmcq. Both are wrong. It’s seventy three million five thousand and twelve and a half.Antonboat (talk) 13:02, 2 January 2018 (UTC)

Why the infinite multitude of nines of 0.999... can’t have an infinite number.

I think I found the following fundamental ‘soft spots’ in Cantor’s theory of infinity:

1) He didn’t proof that the set {0, 1, 2, 3, etc.} could be an actual infinite set: a set that really could have all the natural numbers, none excepted. (According to: https://en.wikipedia.org/Axiom_of_infinity, sub: “Interpretation and consequences”, the existence of such a set is even now not unambigously demonstrated.) Euclid’s Theorem, https://en.wikipedia.org/Euclid%27s_theorem, proves that there cannot be a largest prime. As primes are natural numbers, so there cannot be a largest natural too. And if there can’t be a largest natural, the only actual possible infinite set of naturals must be an unbounded set of naturals, that, being unbounded, can impossibly actually have all the naturals, none excepted.

2) He didn’t define the infinite numbers in the sense that he didn’t indicate the exact spot on a number line where, as successor of a natural, not a natural but an infinite number could appear on it. (Even now, such an unambiguous definition of an infinite number is still not given in: https://en.wikipedia.org/Aleph_number, sub: Aleph number.) But Cantor was convinced that infinite numbers could really exist because he thought he had proved this his diagonal argument. But can they really exist?

3) The conclusion he came to by interpreting the result he found in his diagonal argument can impossibly be true.

On the site: https://en.wikipedia.org/Cantor%27s_diagonal_argument, in: “Uncountable set” is written: “In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one). He begins with a constructive proof of the following theorem: If ${\displaystyle s_{1}}$, ${\displaystyle s_{2}}$, … , ${\displaystyle s_{n}}$, … is any enumeration of elements from T, then there is always an element ${\displaystyle s}$ of T which corresponds to no ${\displaystyle s_{n}}$ in the enumeration. By construction, ${\displaystyle s}$ differs from each ${\displaystyle s_{n}}$, since their ${\displaystyle n}$th digits differ (highlighted in the example). ((As I expect you to be familiar with Cantor’s way of working, I omitted that example here, Antonboat)) Hence, ${\displaystyle s}$ cannot occur in the enumeration. Based on this theorem, Cantor then uses a proof by contradiction to show that: The set T is uncountable. He assumes for contradiction that T was countable. Then all its elements could be written as an enumeration ${\displaystyle s_{1}}$, ${\displaystyle s_{2}}$, … , ${\displaystyle s_{n}}$, …. Applying the previous theorem to this enumeration would produce a sequence ${\displaystyle s}$ not belonging to the enumeration. However, ${\displaystyle s}$ was an element of T and should therefore be in the enumeration. This contradicts the original assumption, so T must be uncountable.“

But is this conclusion true? In my opinion it cannot be true. And that can easily be shown. One may write the following paraphrase of the proof above in digits, instead of the binary digits Cantor used. Suppose that an infinite list could be made in which each positive integer ${\displaystyle n}$ is matched up with, in random order, a real number ${\displaystyle r(n)}$ between 0 and 1, the reals given by infinite decimals. The first five lines of this list might look then as follows:

${\displaystyle r(1)}$ 0.44159...

${\displaystyle r(2)}$ 0.34333...

${\displaystyle r(3)}$ 0.71428...

${\displaystyle r(4)}$ 0.41441...

${\displaystyle r(5)}$ 0.50004...

... ...

The digits that run down the diagonal in underlined boldface are: 4, 4, 4, 4, 4, …. By subtracting 1 from these digits, a number: 0.33333... is found. This number is not in the here given list of real numbers.

Sub: ‘Interpretation’ in Misplaced Pages Cantor's diagonal argument, https://en.wikipedia.org/Cantor%27s_diagonal_argument, is said: “In the context of classical mathematics, … the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros (read here for ones and zeros: digits, Antonboat) than there are natural numbers.”

But can this interpretation be right indeed? Every mathematician knows that an actual infinite list of reals < 1 has to contain reals like: 0.44444…, 0.33333…, 0.999…, 0.5, 0.9…, etc. The list of reals < 1 above is interpreted as an infinite list of all the reals < 1, none excepted. But looking at it we see in reality a list containing only 5 elements, thus a list in which the group of the still remaining infinite multitude of possible reals < 1 not is included. That’s the reason that the list of reals < 1 above, can impossible be an actual infinite enumeration of the reals < 1. And that’s why the in the classical mathematics conventional interpretation of the result of this argument: that it proofs that there are actually more reals < 1 than there are natural numbers, can’t be true. In my opinion the result found by means of the diagonal argument, must be interpreted as a definite proof that the reals < 1 impossibly could be enumerated in an actual infinite and complete list. In that way the diagonal argument proofs for the reals < 1, what Euclid’s Theorem proved for the primes and thus the naturals.

From these proofs together follows that the assumption of the existence of an actual infinite set, be it naturals or reals, limited by ‘infinite’, must be illogical. And if there can’t be a limit ‘infinite’ there can’t be infinite numbers. (In that case the reals < 1, enumerated in random order: 0.99, 0.9, 0.9999, 0.999, etc, can boundlessly be paired with the enumeration: 0, 1, 2, 3, etc, in this way: , , , .) And the nine in the notation of the repeating nine: 0.(9) must then be interpreted as a bounded series of one nine, to which boundlessly is added a new nine. This infinite process of boundlessly adding can’t have a limit ‘infinite’ because of its boundlessness. So any possible number of nines, (and naturals etc.) can only be expressed in a natural number. The only way that remains then to interpret and define ‘infinite’ could be: defining ‘infinite’ as a process of unbounded increase or decrease (in number or length), or as a process of boundlessly continuing, (of time).Antonboat (talk) 13:12, 2 January 2018 (UTC)

A good and happy 2018 to you all!˜˜˜˜ — Preceding unsigned comment added by Antonboat (talk • contribs) 10:32, 2 January 2018 (UTC)

@Antonboat: Wrong. You must start with fundamentals like the rigorous definition of an infinite set. The rest of your post is the same hand-waving that got you nowhere before Christmas. If you are rejecting the axiom of infinity then you cannot possibly be talking about the natural numbers, an object whose existence is the axiom of infinity.--Jasper Deng (talk) 10:38, 2 January 2018 (UTC)

Dear Jasper Deng You write: ”If you are rejecting the axiom of infinity then you cannot possibly be talking about the natural numbers, an object whose existence is the axiom of infinity.” Before there was an axiom of infinity there was Euclides and thus there were primes/ natural numbers.Antonboat (talk) 13:07, 2 January 2018 (UTC)

@Antonboat: What came before the axiomization of mathematics is not sufficiently rigorous. Why are you so resistant to embracing the rigor I expect of you for a discussion like this?--Jasper Deng (talk) 18:08, 2 January 2018 (UTC)

Dear Jasper Deng. You told me I had to be more rigorous in my reasoning. So I tried this in this note by citing Misplaced Pages. Apparently not to your full satisfaction. But is your comment a substantive comment? I’m afraid it is not. In my opinion, you’re barking again without biting. An axiom is as far as I know a hypothesis and not a rigorous proof.

Show to me by a rigourous proof that there can be a infinite set of all the naturals, none excepted.

Show me the exact spot on a number line where naturals change in infinite numbers.

Show me that I’m wrong in my expecting that reals like 0.3333, 0.9 and 0.314 should be in in a set of which is assumed that it can be the set of all reals< 1, none excepted.

Assuming you to be an able mathematician, I expect from you brainy answers, not any rudeness. Or am I asking too much? In that case please: "Sit!" and: "Be still!".Antonboat (talk) 07:19, 4 January 2018 (UTC)

I already told you: the existence of the natural numbers as an infinite set is a fundamental axiom of ZFC set theory and as such, cannot be proven from the other axioms of that theory; in fact, the definition of "finite" and "infinite" relies on the existence of that set so this is pretty much by definition. To prove it requires assuming something at least as strong as that. Merely "citing" Misplaced Pages does not count as "rigor". Refer to Diego's comments on this page for what I mean by it. And there is no "exact spot on a number line where naturals change in infinite numbers", because the infinite ordinals are not contained in the natural numbers and a number line represents only the set of natural numbers.--Jasper Deng (talk) 07:23, 4 January 2018 (UTC)

(edit conflict)Jasper, please don't bite the newbies. If Antonboat does not have a formal knowledge of the axiomatization of real numbers, you should participate in the discussion by explaining how he can get it and pointing out some good books he can read to acquire it, not complaining that he doesn't use a theory that he doesn't know about. Diego (talk) 23:09, 2 January 2018 (UTC)

Dear Diego.Thank you for your understanding of my position in this discussion. But I've learned to accept real truths although they may be hard to swallow. For, although I may be a newby in your eyes, I'm absolute not a baby. I want to know as we say in Dutch: "het naadje van de kous" (literally translated: "the seam of the stocking"), by which I mean that I should like to know the smallest fundamental details of something, in this case the rigorous proofs, not axiomata!, a mathematician could give me as an answer to the questions I asked Jasper Deng above.Antonboat (talk) 07:51, 4 January 2018 (UTC)

@Diego Moya: I don't think I need to be any less blunt when I have pointed him to in the right direction something like a dozen times above, including links to the relevant articles, and detailed explanations of how to approach this topic. I'm emphatically not an intuitionist and usually don't believe that's the best pedagogical approach. I also do not believe that providing him with a list of books (or other offline resource) is going to be very useful for him, as he would have to go out of his way to obtain those books.--Jasper Deng (talk) 08:11, 3 January 2018 (UTC)

That argument reminds me a bit of Giovanni Girolamo Saccheri saying when trying to prove the parallel postulate "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". And Saccheri actually did know some mathematics. The modern requirements of rigor have meant that Euclid's axioms have had to be augmented with a whole load of new ones - you might like to have a look at Hilbert's axioms and then Tarski's axioms to see what is meant by that. In them one could replace the word point by quib and line by walb etc and the arguments would still hold and give valid theorems. The natural numbers and infinity are like that in modern mathematics. Dmcq (talk) 23:06, 2 January 2018 (UTC)

Hello Antonboat. "Newbie" is a term we use for new editors of Misplaced Pages; it speaks only about your exposition to the community norms and customs, not your life experience. Your account information tells me that you've made just 48 edits in total starting last december, which makes you a newcomer in this sense.

From your words I think you have a confusion regarding the role of axioms and hypotheses in logic and maths; both are logic assertions, but they serve different purposes. I'll try to provide a very brief explanation; it's possible that you already have some knowledge about the following, but I think this will help explain a misscomunication that's going on in this conversation.

• In formal logic, a theory is a well-defined set of (umambiguous) logic assertions called axioms, plus all the logic assertions ("theorems") that can derived from that set using inference rules.

-If one of the axioms in the original set can be derived from the others, you can remove it from the set and get an equivalent but simpler theory. So in practice, you can assume that all the axioms in a theory are independent from each other, and all are needed to define the theory.

- If you add a new independent axiom to the original set, you get a new theory, different from the previous one, which may have different logical consequences.

• A theory is consistent if it is impossible to derive a contradiction from the set of axioms using the allowed inference rules.
• A hypothesis is a logic assertion that you are trying to derive from the set of axioms in the theory. If you can find that derivation, the hypothesis is proven and becomes a theorem. But if you create a theory containing the axioms and the hypothesis, and you can derive a contradiction from that, the hypothesis is disproven.

In this conversation, you're treating the existence of infinite sets as a hypothesis that needs to be proved (or disproved) from the axioms of natural and real numbers. However, this happens to be an unreasonable request, since it is known that it is impossible to create such proof from those axioms; and also that it is impossible to disprove the existence of infinite sets from them: the theory containing the axioms of the reals plus the existence of infinite sets is consistent. (By the way, your attempt at a disproof is not precise enough to count as a formal logic deduction). This existence or inexistence of infinite sets is not a theorem in the theory of naturals and reals, i.e. not an assertion that can be derived from their axioms. So, when mathematicians use infinite sets, they do it as a new axiom, because their existence is an independent logic assertion, and therefore it can only be treated as an independent axiom that expands the basic number theory.

I hope this helps. I'm sure the other editors will correct me if I've made some mistake or over-simplification somewhere. ;-) Diego (talk) 10:08, 4 January 2018 (UTC)

Dear Diego Moya. I hope you didn’t think I felt offended by you naming me “newby”. Yes I can’t deny it, I’m a newcomer in your world. Thank you for your detailed explanation of the different use of the term axiom in logic and mathematics. But no, I don’t have a confusion regarding the role of axioms and hypotheses in logic and maths. What I miss in your overview is the description of the qualitative difference that exists between both types of axiom. In: https://en.wikipedia.org/Axiom, sub Axiom, is written: “ Logical axioms are usually statements that are taken to be true within the system of logic they define ... . ... non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory.” So a non-logical axiom like the axiom of infinity, is an assumption like a hypothesis.

You write: “In this conversation, you're treating the existence of infinite sets as a hypothesis that needs to be proved (or disproved) from the axioms of natural and real numbers. “ Yes, you’re right. And one rigorous proven axiom in your mathematics is Euclid’s Theorem that concludes that there can’t be a largest prime and following him I conclude thereafter: and thus no largest natural. You write of this reasoning: “(By the way, your attempt at a disproof is not precise enough to count as a formal logic deduction).” Is the fact that every n + 1 > n not proof enough that there can’t be a set/list, or whatever name a mathematician may give to it, of the infinite multitude of natural numbers, non excepted? Why than accept that Cantor proofs in his diagonal argument that a list/set of all the reals < 1 is impossible?

And especially: why than accept the axiom of infinity as the real truth, knowing Euclid’s Theorem?Antonboat (talk) 09:29, 8 January 2018 (UTC)

@Antonboat: Why can't a set be of infinite cardinality?--Jasper Deng (talk) 09:37, 8 January 2018 (UTC)

Thank you, Diego Moya, for this nice summing up, I truly hope that this covers a good deal of the accumulating misunderstandings. There is just one detail I am a bit weary of. To my knowledge there are several axiomatic systems to specifically establish "natural numbers and their arithmetic", even in slightly different expressive power, but all of them contain an axiom for the existence of the "countable" infinity (induction?). I am not sufficiently educated to be similarly explicit about the various axiomatic systems for "real numbers" and for "sets" in the various settings (ZF(C), Gödel, ..., transfinite induction?), that all do allow for higher infinities, and that all are capable of implementing the natural numbers.

Besides this, one should also note, that extracting countable lists from the real numbers, to show that there are only countably many such numbers, is begging the question, and that Cantors 'list', spreading infinitely in two directions, which is explicitly conceived and assumed to exhaust all real numbers, is shown to fail this task by not containing at least(!) one other real number, constructed as different to all the countably infinitely many numbers in the list, thereby concluding the proof by contradiction (reductio ad absurdum) of the non-existence of such an exhaustive and countable list. (Remark: this diagonalization argument is somewhat different to the diagonalization argument of proving the countability of rational numbers, also involving a 2-dim list)

Perhaps, it helps to mention additionally that there exists a (small?) fraction of mathematicians, that does not accept these axiomatic systems as a valid basis of mathematical work, some of them additionally discarding parts of the rules for the mentioned derivation of theorems from the axioms, which the vast majority of mathematicians gracefully embrace. Purgy (talk) 11:08, 4 January 2018 (UTC)

How to number any boundless and well ordered enumeration of reals smaller and bigger than one with naturals.

How could these reals be numbered, presented in a well ordered way? For the naturals this is simple because there is a smallest natural. That’s why numbering the naturals of the unbounded well ordered enumeration of naturals: 0, 1, 2, 3, etc with ordinals 0, 1, 2, 3, etc., is not difficult: pairing both enumerations this way:

, , , , .

But a well ordered unbounded enumeration starting from the smallest real, is impossible. The reason for this is that for every real number 10E-n we choose as starting point for an enumeration like: 1.10E-n, 2.10E-n, 3.10E-n, we could choose an infinitely smaller one. Where to begin with numbering a well ordered unbounded series of reals? Starting from 1 = 1.10E0 as the supposed smallest real, reals could be paired with ordinals in this way:

, , , .

Starting from 1.10E-1 as the supposed smallest number the pairing:

, , , is found.

And in general, starting from any 1.10E-n as the supposed smallest real the pairing :

, , , is found.

For any n applies: if one assigns the ordinal 1 to a real 10E-n, all the reals in the unbounded well ordered enumeration of reals: 1.10E-n, 2.10E-n, 3.10E-n, etc, could be paired to a unique ordinal which can be found by multiplying each real with 10En. As an example the following.The repeating nine 0.(9) may be seen as represented by the unbounded addition: 0.9 + 0.09 + 0.009 + 0.0009, etc. As the first element of this addition is a real: 9.10E-1, the nth element must be a real 9.10E-n. Starting from 10E-4 as the supposed smallest real, 0.0009 must be paired to the ordinal: 0.0009 x 9. 10E4 = 9, 0.009 must be paired to the ordinal 0.009 x 10E4 = 90, 0.09 must be paired to the ordinal: 0.09 x 10E4 = 900 and 0.9 must than be paired to: 0.9 x 10E4 = 9000. Applying this rule then to the real 9999 this real/natural must be numbered with 99999.10E4 = 99,990,000. The only exception to this rule is 0. If 0 is seen as the smallest real, applying this rule leads to the result that only an ordinal 1 could be assigned as shown here: , , 3.0 = 0, 1], .Antonboat (talk) 06:54, 4 January 2018 (UTC)

You are running into the motivation for having the axiom of choice, which has been needed to show the existence of a well-ordering of the reals. There is no smallest real number, however, just like there is no biggest real number. Be careful with trying to envision infinite sets as infinite "lists". For the real numbers, such a listing does not exist when the indexing set is the natural numbers.--Jasper Deng (talk) 07:11, 4 January 2018 (UTC)

Dear Jasper Deng. You write: “You are running into the motivation for having the axiom of choice, which has been needed to show the existence of a well-ordering of the reals.” Why? In: https://en.wikipedia.org/Well-orderIn mathematics, is said: “a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. Following this definition I stated that: 1.10E-1, 2.10E-1, 3.10E-1, etc, is a well ordered enumeration. What’s wrong with that? You write: “Be careful with trying to envision infinite sets as infinite "lists". For the real numbers, such a listing does not exist when the indexing set is the natural numbers.” By “infinite list/enumeration” I meant: a bounded list of reals to which bounlessly are added new unique real numbers being multiple negative powers of ten. Following my interpretation of infinite it can be done, for example: , , , .Antonboat (talk) 10:09, 8 January 2018 (UTC)

@Antonboat: No, that is not a well-ordering of all the real numbers. Where does a number like 1/9 = 0.11111111..., pi, or even 0.125 fit in your scheme? Again, you are doing yourself no favors by thinking of infinite sets as "a bounded list of reals to which boundlessly are added new unique real numbers". When you say a "list" or "listing", you must identify an index set in order to make a meaningful (rigorous) statement.--Jasper Deng (talk) 10:16, 8 January 2018 (UTC)

Dear Jasper Deng. It seems you comments faster that I can respond to it. You write: “@Antonboat: No, that is not a well-ordering of all the real numbers. “ You’re right, but, like you, I believe in Cantor’s diagonal argument which proved that you can’t have all the reals, none excepted, in a set/list/enumeration. And in my note I didn;’t pretend that. What I said there was: “For any n applies: if one assigns the ordinal 1 to a real 10E-n, all the reals in the unbounded well ordered enumeration of reals: 1.10E-n, 2.10E-n, 3.10E-n, etc, could be paired to a unique ordinal which can be found by multiplying each real with 10En.” You ask me: “Where does a number like 1/9 = 0.11111111..., pi, or even 0.125 fit in your scheme?” Please see about this my answer to Dmcq, Antonboat (talk) 10:37, 8 January 2018 (UTC) You write: “Again, you are doing yourself no favors by thinking of infinite sets as "a bounded list of reals to which boundlessly are added new unique real numbers".” Again: give me a rigorous proof why I may not do that. You write: “When you say a "list" or "listing", you must identify an index set in order to make a meaningful (rigorous) statement.”As far as I know I did this by the index set {1, 2, 3, etc}Antonboat (talk) 11:02, 8 January 2018 (UTC)

A set is not the same as a list, please take out the time to distinguish the two notions. A list, for our purposes, is a surjective function from the index set to the set we are listing, which is not going to be possible when the index set is the natural numbers and the indexed set is the real numbers. I don't need to disprove something that is not a meaningful mathematical statement, as is your "boundlessly are added" notion.--Jasper Deng (talk) 18:13, 8 January 2018 (UTC)

What you are doing is pointless. It is impossible. Even a simple number like 1/3 doesn't fit in your scheme and in general whatever you do it isn't going to work. It has been proved impossible just like squaring the circle using a straight edge and compass, in fact it is more at the level of trying to find a rational number equal to the hypotenuse of a right angle triangles whose two adjacent sides are both 1. Dmcq (talk) 14:37, 4 January 2018 (UTC)

Dear Dmcq. You write: “What you are doing is pointless. It is impossible. Even a simple number like 1/3 doesn't fit in your scheme and in general whatever you do it isn't going to work. It has been proved impossible ... .¨ I’m sorry to say but both a simple number like 1/3 and an difficult number like π fit perfectly in my scheme, witness the following. It's commonly known that neither 1/3 nor π can be assigned an exact value within the decimal system, witness 1 : 3 = 0.3 + 1/30. If it is supposed that 10E-4 is the smallest real, applying the rule given in my note for 1/3 the ordinal 0.3333 x 10,000 = 3333 is found and for π the ordinal 3,1415 x 10,000 = 31415. For 3333 we find than the ordinal: 33,330,000, for 33,330,000 the ordinal: 333,300,000,000, etc. Supposing 10E-n is the smallest real and n being 100,000 we find the ordinal : 10E-100,000 x 10E-100,000 = 1.Antonboat (talk) 10:37, 8 January 2018 (UTC)

@Antonboat: Sorry, but you have not defined a function. There is no "smallest" real, so you are not free to choose one. Even disregarding that, your scheme is not one-to-one: 0.3333444... (4/9 - 1000/9999) and 0.3333... (1/3) get assigned the same natural number for the "smallest real" 10 and for every "smallest real" 10 there always exists two rational (in particular, real) numbers which get assigned the same natural number. Please read the definition of a well-ordering closely.--Jasper Deng (talk) 10:44, 8 January 2018 (UTC)

Dear Jasper Deng. You write: “There is no "smallest" real, so you are not free to choose one.” Who or what forbids me to do that? As a mathematician you feel free to assume an infinite set of naturals, none exempted. You write: “Even disregarding that, your scheme is not one-to-one: 0.3333444... (4/9 - 1000/9999) and 0.3333... (1/3) get assigned the same natural number for the "smallest real" 10-4 ... .” When I choose 10E0 then 3 and π have the same ordinal: 3, but not assuming 10E-1 is the smallest real. Then 3 has the ordinal 30 and π the ordinal 31. If you choose 10E-5 as smallest real 0.3333444... has the ordinal number 33334 and 0.3333... the ordinal number 3333.

You write “... and for every "smallest real" 10-n there always exists two rational (in particular, real) numbers which get assigned the same natural number.0 as the smallest real have the same ordinal. “ Please tell me more about this.Antonboat (talk) 11:34, 8 January 2018 (UTC)Antonboat (talk) 11:39, 8 January 2018 (UTC)

@Antonboat: You can't pick any "smallest real" such that your scheme enumerates even all the rational numbers, let alone all the real numbers. If you choose 10 then similarly 0.3333344... and 0.333333... are a contradicting pair. I put "smallest real" in quotes because there exists no such object, so you are talking about a nonexistent object.--Jasper Deng (talk) 18:13, 8 January 2018 (UTC)

Dear Jasper Deng. You write: “I put "smallest real" in quotes because there exists no such object, so you are talking about a nonexistent object.” Zermelo and Fraenkel assumed that there could be an infinite set of all the naturals, incidentally without proving that! So why may I not suppose that 10E-5 is the smallest real in my reasoning, explicite stating therein that this assumption can’t be true? Because I’m not a mathematician? You write: “@Antonboat: You can't pick any "smallest real" such that your scheme enumerates even all the rational numbers, let alone all the real numbers.“ a) My dear Jasper, give me an exact decimal value for 1/3 and I’ll give you an exact natural to number it. b) I never spoke of all the reals, but of all the reals one can create. You write: “If you choose 10-5 then similarly 0.3333344... and 0.333333... are a contradicting pair.“ No my dear Jasper, starting from 10E-5 there can’t be numbers like 0.3333344... and 0.333333... Only a number 0.33333. If you want to pair the exact decimal values 0.3333344 and 0.333333 with natural numbers you’ll have to start numbering from 10E-7 and then find the ordinals 3333344 and 3333333 for them. And of course you know that numbers like 0.3333344... and 0.333333... are not defined in the decimal system. And we’re arguing about the numbering of reals belonging to the decimal system 10E-n to 10En , any n being < n + 1.Antonboat (talk) 08:00, 9 January 2018 (UTC)

@Antonboat: Sorry, this is once again not even wrong. None of your proposed "numberings" covers even an infinite subset of the reals, let alone all of them. A function can assign one and only one value to each member of its domain. So you are not free to keep moving the goalposts by changing your value of n. Doing so produces a different enumeration and is not valid for all the reals - no finite value of n will suffice to cover all real numbers. For what it's worth, the set of real numbers with finite decimal expansion is countable, though not using what you propose (which does not even cover that subset of reals). But almost all real numbers have infinite decimal expansions.--Jasper Deng (talk) 08:05, 9 January 2018 (UTC)

Dear Jasper Deng "Not even wrong" is barking, not biting. And: "Sticks and stones may break my bones but you could never hurt me" You write: “None of your proposed "numberings" covers even an infinite subset of the reals, let alone all of them. “ Again, for I’ve asked you many times before, give me a rigorous proof that an infinite set of all the reals, none excepted is possible. In my comments on Cantor’s diagonal argument I showed that, starting from bounded sets of reals and naturals he concluded that an infinite set of all the reals < 1 could not be possible, because this argument showed the possibility of creating a boundless multitude of reals. You write: “A function can assign one and only one value to each member of its domain.“ you’re right. You write: “So you are not free to keep moving the goalposts by changing your value of n. “ In my opinion You’re wrong here. If I choose a new domain I may choose a new value for n. You write: “Doing so produces a different enumeration and is not valid for all the reals ...” You’re right it’s only valid for the domain that I chose. Please see also about this my new section: Why Cantor couldn’t succeed in his effort to number the infinite multitude of reals with natural numbers¨ You write: “ But almost all real numbers have infinite decimal expansions.”It’s a pity but then they have no exact decimal value and so can’t be numbered by a natural belonging to the decimal system.Antonboat (talk) 09:17, 9 January 2018 (UTC)

@Antonboat: I'm afraid I'm going to have to be frank here, by restating what I said above. If you find yourself unable to grasp fundamentals like the rigorous definition of infinity and how to use rigor in mathematical discussion, we won't have this conversation.--Jasper Deng (talk) 09:21, 9 January 2018 (UTC)

Why Cantor couldn’t succeed in his effort to number the infinite multitude of reals with natural numbers

Cantor believed he had proved that there could not be naturals left after numbering all the possible reals smaller than one. It is quite understandable that he came to this conclusion. Let’s suppose that 10E-n is the smallest real in the decimal system. Then 10En must be the largest real in this system. Starting from 1.10E-n as the smallest real, we must conclude that the enumeration 1.10E-n, 2.10E-n, 3.10E-n must stop when we reach 10E-n.10E-n = 1. By supposing that 10En is the largest real there can be no naturals left to number the reals ≥ 1. That’s why he had to come to the conclusion that the infinite number of reals had to be infinite larger than the infinite of the reals. From it he deduced: as there are no naturals left to number the real 1 + 10E-n there must be numbers larger than any natural number: infinite numbers. A great find, found by a great and daring explorer. But could he be right in thinking this way? Let us see. Cantor proved in his diagonal argument that starting from a bounded enumeration of reals < 1 there could boundlessly be reals < 1 added to this enumeration. Euclid proved the same for the primes, thus for the naturals, using the principle n + 1 > n. That’s why for every real we suppose to be the smallest real, there is an infinite multitude of reals smaller than that. That’s why for every real/ natural we suppose to be the largest real/natural there is an infinity of reals/naturals larger than that. And in that case we can number the real 1 + 10E-n with a natural: 10En + 1, the real 2 with 2.10En, the real 10nEn with 10nEn.10nEn , the real 1,000,000 with: 1,000,000.10En and so on. And Euclid’s theorem proved for the primes, thus naturals, that it is impossible to exhaust the for this numbering available infinite multitude of naturals. Challenge to the reader: show me by a brainy reasoning that I'm wrong here.Antonboat (talk) 07:23, 9 January 2018 (UTC)

You say: "... {nonsense} ... But could he be right in thinking this way?" Cantor never thought this way. Mathematicians don't think this way . - DVdm (talk) 07:47, 9 January 2018 (UTC)

Dear DVdm. a) Thank you for your evaluation of my note. I invite you here to give a brainier than "nonsense". b) Can you prove that Cantor has never thought this way. Best regards.Antonboat (talk) 08:25, 9 January 2018 (UTC)

Nonsense is nonsense. It can't get brainier than "nonsense". Sorry. - DVdm (talk) 08:51, 9 January 2018 (UTC)

@DVdm: I'm wondering if I am being productive attempting to explain this to Antonboat. Obviously, I'm somewhat frustrated that he doesn't want to start with first principles.--Jasper Deng (talk) 08:54, 9 January 2018 (UTC)

@Jasper Deng: it is clear to me that everyone is wasting their time here. And frankly, I'm reluctant to be completely frank... - DVdm (talk) 09:04, 9 January 2018 (UTC)

@DVdm: I mean, I don't want User:Antonboat to feel bad, but I was just wondering if I was the only one feeling that way, since Antonboat is simply not asking the right questions, a prerequisite for learning math.--Jasper Deng (talk) 09:08, 9 January 2018 (UTC)

I don't think we can help him ask the right questions. I think we must say sorry, please go elsewhere, nobody can help you here. But this is getting off-topic (in an off-topic part of Misplaced Pages to begin with ) - DVdm (talk) 09:16, 9 January 2018 (UTC)

@DVdm: Beat you to it (see my previous edit).--Jasper Deng (talk) 09:18, 9 January 2018 (UTC)

@Antonboat: "Cantor believed he had proved that there could not be naturals left after numbering all the possible reals smaller than one." - please state exactly what you mean by this in a well-defined manner. To better understand the subtleties of different infinite cardinalities, read Hilbert's paradox of the grand hotel. The rest of your comment is a complete misunderstanding of Cantor's argument.--Jasper Deng (talk) 07:57, 9 January 2018 (UTC)

Dear Jasper Deng. You wrote to me on 09:49, 23 December 2017 (UTC) “The set of real numbers between 0 and 1 exclusive is a bounded subset of the real numbers and yet is of infinite cardinality.”Antonboat (talk) 08:26, 9 January 2018 (UTC)

Okay and how does that relate to what you said?--Jasper Deng (talk) 08:27, 9 January 2018 (UTC)

My, but possibly not Cantor's, way of reasoning is: "By supposing that 10En is the largest real there can be no naturals left to number the reals ≥ 1.¨Antonboat (talk) 08:38, 9 January 2018 (UTC)

@Antonboat: This is not a meaningful statement. What does it mean to have "naturals left"? To "number the reals"? You are technically not working with the real numbers themselves when you claim there is a "largest" real, so I have been understanding you as working with a subset of the reals. But even then, your argument fails and is not even close to what Cantor is saying. What Cantor is saying is that there is no surjective function from the natural numbers to the reals, or even an open interval within the reals - i.e. there is no "numbering" to speak of in the first place. And again, please drop your conception of "boundlessly adding", per what I said above. I would also repeat what Purgy said above -- wean yourself off of decimal notation for real numbers. It seems to be blinding you to the fundamentals of a construction of the real numbers, which makes no appeal to decimal expansions at all.--Jasper Deng (talk) 08:43, 9 January 2018 (UTC)