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End extension

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Extension of a transitive set

In model theory and set theory, which are disciplines within mathematics, a model B = ⟨ B , F ⟩ {\displaystyle {\mathfrak {B}}=\langle B,F\rangle } of some axiom system of set theory T {\displaystyle T} in the language of set theory is an end extension of A = ⟨ A , E ⟩ {\displaystyle {\mathfrak {A}}=\langle A,E\rangle } , in symbols A ⊆ end B {\displaystyle {\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}} , if

  1. A {\displaystyle {\mathfrak {A}}} is a substructure of B {\displaystyle {\mathfrak {B}}} , (i.e., A ⊆ B {\displaystyle A\subseteq B} and E = F | A {\displaystyle E=F|_{A}} ), and
  2. b ∈ A {\displaystyle b\in A} whenever a ∈ A {\displaystyle a\in A} and b F a {\displaystyle bFa} hold, i.e., no new elements are added by B {\displaystyle {\mathfrak {B}}} to the elements of A {\displaystyle A} .

The second condition can be equivalently written as { b ∈ A : b E a } = { b ∈ B : b F a } {\displaystyle \{b\in A:bEa\}=\{b\in B:bFa\}} for all a ∈ A {\displaystyle a\in A} .

For example, ⟨ B , ∈ ⟩ {\displaystyle \langle B,\in \rangle } is an end extension of ⟨ A , ∈ ⟩ {\displaystyle \langle A,\in \rangle } if A {\displaystyle A} and B {\displaystyle B} are transitive sets, and A ⊆ B {\displaystyle A\subseteq B} .

A related concept is that of a top extension (also known as rank extension), where a model B = ⟨ B , F ⟩ {\displaystyle {\mathfrak {B}}=\langle B,F\rangle } is a top extension of a model A = ⟨ A , E ⟩ {\displaystyle {\mathfrak {A}}=\langle A,E\rangle } if A ⊆ end B {\displaystyle {\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}} and for all a ∈ A {\displaystyle a\in A} and b ∈ B ∖ A {\displaystyle b\in B\setminus A} , we have r a n k ( b ) > r a n k ( a ) {\displaystyle rank(b)>rank(a)} , where r a n k ( ⋅ ) {\displaystyle rank(\cdot )} denotes the rank of a set.

Existence

Keisler and Morley showed that every countable model of ZF has an end extension which is also an elementary extension. If the elementarity requirement is weakened to being elementary for formulae that are Σ n {\displaystyle \Sigma _{n}} on the Lévy hierarchy, every countable structure in which Σ n {\displaystyle \Sigma _{n}} -collection holds has a Σ n {\displaystyle \Sigma _{n}} -elementary end extension.

References

  1. H. J. Keisler, J. H. Silver, "End Extensions of Models of Set Theory", p.177. In Axiomatic Set Theory, Part 1 (1971), Proceedings of Symposia in Pure Mathematics, Dana Scott, editor.
  2. Keisler, H. Jerome; Morley, Michael (1968), "Elementary extensions of models of set theory", Israel Journal of Mathematics, 5: 49–65, doi:10.1007/BF02771605
  3. Kaufmann, Matt (1981), "On existence of Σn end extensions", Logic Year 1979–80, Lecture Notes in Mathematics, vol. 859, pp. 92–103, doi:10.1007/BFb0090942, ISBN 3-540-10708-8
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