Misplaced Pages

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article. When placing this tag, consider associating this request with a WikiProject. (January 2025) The topic of this article may not meet Misplaced Pages's general notability guideline. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted. Find sources: "Walks on ordinals" – news · newspapers · books · scholar · JSTOR (January 2025) (Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (January 2025) (Learn how and when to remove this message) This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (January 2025) (Learn how and when to remove this message) (Learn how and when to remove this message)

In mathematics, the method of walks on ordinals, often called minimal walks on ordinals, was invented and introduced by Stevo Todorčević as a part of his new proof of the existence of the Countryman line in May 1984.

The method is a tool for constructing uncountable objects (from set-theoretic trees to separable Banach spaces) by utilizing an analysis of certain descending sequences of ordinals known as minimal walks.

The method might be formally described this as follows:

Let ${\displaystyle \alpha }$ be an ordinal and let ${\displaystyle \beta }$ be another one defined as ${\displaystyle \beta =\{\alpha :\alpha <\beta \}}$

Further

${\displaystyle {\begin{aligned}&0=\emptyset ;\\&1=\{0\};\\&2=\{0,1\};\\&\,\,\,\vdots \\&\omega =\{0,1,2,\ldots \};\\&\omega +1=\omega \cup \{\omega \};\\&\omega +2=\omega \cup \{\omega ,\omega +1\};\\&\,\,\,\vdots \end{aligned}}}$

and the Cantor's normal form of an ordinal is ${\displaystyle \alpha =n_{1}\omega ^{\alpha _{1}}+n_{2}\omega ^{\alpha _{2}}+\cdots +n_{k}\omega ^{\alpha _{k}}}$ where ${\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{k}\geq 0}$ are ordinals and ${\displaystyle n_{1},n_{2},\ldots ,n_{k}}$ are natural numbers. For more details see Ordinal arithmetic.

Ordinals from the class ${\displaystyle \varepsilon _{0}=\{\alpha :\alpha =\omega ^{\alpha }\}}$ – in this case the ordinals ${\displaystyle \alpha _{1}>\alpha _{2}>\cdots >\alpha _{k}}$ are all in the Cantor normal form of ${\displaystyle \alpha }$ and smaller than ${\displaystyle \alpha }$. For each limit countable ordinal ${\displaystyle \alpha <\varepsilon _{0}}$ we'll create a sequence ${\displaystyle C_{\alpha }=\{c_{\alpha }(n):\alpha <\omega \}\subseteq \alpha }$ such that ${\displaystyle c_{\alpha +1}(n)=\alpha }$ for all ${\displaystyle n}$ and such that ${\displaystyle c_{\alpha }(n)<c_{\alpha }(n+1)}$ for all ${\displaystyle n}$ and ${\displaystyle \alpha =\lim _{n\to \infty }c_{\alpha }(n)}$ when ${\displaystyle \alpha }$ is limit.

Minimal step from ${\displaystyle \beta }$ towards ${\displaystyle \alpha <\beta }$

${\displaystyle \beta \curvearrowright c_{\beta }(n(\alpha ,\beta ))}$

where

${\displaystyle n(\alpha ,\beta )=\min\{n:c_{\beta }(n)\geq \alpha \}}$

Minimal walk from ${\displaystyle \beta }$ towards ${\displaystyle \alpha }$ is a finite decreasing sequence

${\displaystyle \beta =\beta _{0}\curvearrowright \beta _{1}\curvearrowright \cdots \curvearrowright \beta _{k}=\alpha }$

such that for all ${\displaystyle i<k}$ the step ${\displaystyle \beta _{i}\curvearrowright \beta _{i-1}}$ is the minimal step from ${\displaystyle \beta _{i}}$ towards ${\displaystyle \alpha }$ i.e.

${\displaystyle \beta _{i+1}=c_{\beta }(n(\alpha ,\beta _{i}))}$

The method definition given above belongs to Todorcevic. Different, but equivalent method definitions, can be found in papers.

Many applications of the method have been found in combinatorial set theory, in general topology and in Banach space theory.

Method characteristics

This section needs expansion. You can help by adding to it. (January 2025)

Method use

This section needs expansion. You can help by adding to it. (January 2025)

Sources

• Todorčević, Stevo (1987), "Partitioning pairs of countable ordinals", Acta Mathematica, 159 (3–4): 261–294, doi:10.1007/BF02392561, MR 0908147
• Todorčević, Stevo (2007), Walks on ordinals and their characteristics, Progress in Mathematics, vol. 263, Basel: Birkhäuser Verlag, ISBN 978-3-7643-8528-6, MR 2355670
• Hudson, William Russell (May 2007), Minimal walks and Countryman lines, MSc thesis, Boise State University
• Rinot, Assaf (2012), "Transforming rectangles into squares, with applications to strong colorings", Advances in Mathematics, 231 (2): 1085–1099, doi:10.1016/j.aim.2012.06.013, MR 2955203
• Lücke, Philipp (2017), "Ascending paths and forcings that specialize higher Aronszajn trees", Fundamenta Mathematicae, 239 (1): 51–84, doi:10.4064/fm224-11-2016, MR 3667758

This mathematical logic-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Mathematical logic stubs