Ali Enayat

A standard model of Peano arithmetic with no conservative elementary extension

Abstract The principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion N A ≔ ( N , A ) A ∈ A of the standard model N ≔ ( ω , + , × ) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension N A ∗ = ( ω ∗ , … ) of N A , there is a subset of ω ∗ that is parametrically definable in N A ∗ but whose intersection with ω is not a member of A . We also establish other results that highlight the role of countability in the model theory of arithmetic. Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A / FIN (where FIN is the ideal of finite sets) collapses ℵ 1 when viewed as a notion of forcing.

Automorphisms of models of arithmetic: A unified view

Abstract We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic PA . In particular, we use this method to prove Theoremxa0A below, which confirms a long-standing conjecture of James Schmerl. Theoremxa0A If M is a countable recursively saturated model of PA in which N is a strong cut, then for any M 0 ≺ M there is an automorphism j of M such that the fixed point set of j is isomorphic to M 0 . We also fine-tune a number of classical results. One of our typical results in this direction is Theoremxa0B xa0below, which generalizes a theorem of Kaye–Kossak–Kotlarski (in what follows Aut ( X ) is the automorphism group of the structure X , and Q xa0is the ordered set of rationals). Theoremxa0B Suppose M is a countable recursively saturated model of PA in which N is a strong cut. There is a group embedding j ↦ j ˆ from Aut ( Q ) into Aut ( M ) such that for each j ∈ Aut ( Q ) that is fixed point free, j ˆ moves every undefinable element of M .

Models of set theory with definable ordinals

Abstract.A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the following:1. If T is a consistent completion of ZF+V≠OD, then T has continuum-many countable nonisomorphic Paris models.2. Every countable model of ZFC has a Paris generic extension.3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism.4. For a model ZF, is a prime model ⇒ is a Paris model and satisfies AC⇒ is a minimal model. Moreover, Neither implication reverses assuming Con(ZF).

MODEL THEORY OF THE REGULARITY AND REFLECTION SCHEMES

AbstractThis paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF

Trees and Keislers problem

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Automorphisms of models of bounded arithmetic

(the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here

A recursive nonstandard model for open induction with GCD property and cofinal primes

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