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Largest initial segments pointwise fixed by automorphisms of models of set theory

Enayat, Ali
; 
Kaufmann, Matt
; 
McKenzie, Zachiri
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Gothenburg University; University of Texas at Austin
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2017-09-12
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Mathematics
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Abstract
Given a model M of set theory, and a nontrivial automorphism j of M, let Ifix(j) be the submodel of M whose universe consists of elements m of M such that j(x)=x for every x in the transitive closure of m (where the transitive closure of m is computed within M). Here we study the class C of structures of the form Ifix(j), where the ambient model M satisfies a frugal yet robust fragment of ZFC known as MOST, and j(m)=m whenever m is a finite ordinal in the sense of M. Our main achievement is the calculation of the theory of C as precisely MOST+Δ0P Collection. The following theorems encapsulate our principal results: Theorem A. Every structure inC satisfies MOST+Δ0P Collection. Theorem B. Each of the following three conditions is sufficient for a countable structure (a) N is a transitive model of MOST+Δ0P Collection. (b) N is a recursively saturated model of MOST+Δ0P Collection. (c) N is a model of ZFC. Theorem C. Suppose M is a countable recursively saturated model of ZFC and I is a proper initial segment of OrdM that is closed under exponentiation and contains ωM. There is a group embedding j⟼j from Aut(Q) into Aut(M) such that I is the longest initial segment of OrdM that is pointwise fixed by jˇ for every nontrivial j∈Aut(Q). In Theorem C, Aut(X) is the group of automorphisms of the structure X, and Q is the ordered set of rationals.
Citation
Enayat, A., Kaufmann, M., & McKenzie, Z. (2017). Largest initial segments pointwise fixed by automorphisms of models of set theory. Archive for Mathematical Logic, 57(1-2), 91–139. https://doi.org/10.1007/s00153-017-0582-3
Publisher
Springer
Journal
Archive for Mathematical Logic
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http://hdl.handle.net/10034/629352
DOI
10.1007/s00153-017-0582-3
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Article
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© The Author(s) 2017. This article is an open access publication.
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0933-5846
EISSN
1432-0665
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Unfunded
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https://link.springer.com/article/10.1007/s00153-017-0582-3
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