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Archive for Mathematical Logic

, Volume 57, Issue 1–2, pp 91–139 | Cite as

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

  • Ali EnayatEmail author
  • Matt Kaufmann
  • Zachiri McKenzie
Open Access
Article

Abstract

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results:

Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\).

Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):

(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).

(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).

(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\).

Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\)

In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.

Keywords

Automorphisms Models of set theory Recursively saturated 

Mathematics Subject Classification

Primary 03C62 Secondary 03H9 

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Authors and Affiliations

  1. 1.Department of Philosophy, Linguistics and Theory of ScienceGothenburg UniversityGothenburgSweden
  2. 2.Department of Computer ScienceUniversity of Texas at AustinAustinUSA

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