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BOREL STRUCTURES: A BRIEF SURVEY
"... Abstract. We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. ..."
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Abstract. We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area. 1.
A standard model of Peano arithmetic with no conservative elementary extension, preprint
, 2006
"... Abstract. The principal result of this paper answers a longstanding question in the model theory of arithmetic [KS, 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 ΩA: = (ω, +, ·, X)X∈A of the ..."
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Abstract. The principal result of this paper answers a longstanding question in the model theory of arithmetic [KS, 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 ΩA: = (ω, +, ·, X)X∈A of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension Ω ∗ A = (ω∗, · · ·) of ΩA, there is a subset of ω ∗ that is parametrically definable in Ω ∗ A but whose intersection with ω is not a member of A. Inspired by a recent question of Gitman and Hamkins, we also show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/F IN (where F IN is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing. 1.
ωMODELS OF FINITE SET THEORY
, 2008
"... Abstract. Finite set theory, here denoted ZFfin, is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (ZermeloFraenkel set theory). An ωmodel of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Man ..."
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Abstract. Finite set theory, here denoted ZFfin, is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (ZermeloFraenkel set theory). An ωmodel of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the BernaysRieger method of permutations to construct a recursive ωmodel of ZFfin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ωmodels of ZFfin. In particular, we present a perspicuous method for constructing recursive nonstandard ωmodel of ZFfin without the use of permutations. We then use this method to establish the following central theorem. Theorem A. For every simple graph (A, F), where F is a set of unordered pairs of A, there is an ωmodel M of ZFfin whose universe contains A and which satisfies the following two conditions: (1) There is parameterfree formula ϕ(x, y) such that for all elements a and b of M, M  = ϕ(a, b) iff {a, b} ∈ F; (2) Every element of M is definable in (M, c)c∈A. Theorem A enables us to build a variety of ωmodels with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ωmodel of ZFfin. Corollary 2. For each infinite cardinal κ there are 2 κ rigid nonisomorphic ωmodels of ZFfin of cardinality κ. Corollary 3. There are continuummany nonisomorphic pointwise definable ωmodels of ZFfin.
MODELS OF PA: STANDARD SYTEMS WITHOUT MINIMAL ULTRAFILTERS
, 2010
"... We prove that N has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of N represented in N which is minimal (i.e. Ramsey for partitions represented in N). ..."
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We prove that N has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of N represented in N which is minimal (i.e. Ramsey for partitions represented in N).
MODELS OF PA: WHEN TWO ELEMENTS ARE NECESSARILY ORDER AUTOMORPHIC
"... Abstract. We are interested in the question of how much the order of a nonstandard model of PA can determine the model. In particular, for a model M, we want to characterize the complete types p(x,y) of nonstandard elements (a,b) such that the linear orders {x: x < a} and {x: x < b} are neces ..."
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Abstract. We are interested in the question of how much the order of a nonstandard model of PA can determine the model. In particular, for a model M, we want to characterize the complete types p(x,y) of nonstandard elements (a,b) such that the linear orders {x: x < a} and {x: x < b} are necessarily isomorphic. It is proved that this set includes the complete types p(x,y) such that if the pair (a,b) realizes it (in M) then there is an element c such that for all standard n,c n < a,c n < b,a < bc and b < ac. We prove that this is optimal, because if ♦ℵ1 holds, then there is M of cardinality ℵ1 for which we get equality. We also deal with how much the order in a model of PA may determine the addition. Let M be a model of Peano Arithmetic (PA). For an a ∈ M, by M<a we denote the set {c ∈ M: M  = c < a} with the inherited linear order. For any pair (a,b) of nonstandard elements of M, let (∗)M,a,b be the condition dfeined by
Model Theory of the Reflection Scheme
, 2007
"... This paper develops the model theory of ordered structures that satisfy the analogue, REF (L), of the reflection principle of ZermeloFraenkel set theory. Here L is a language with a distinguished linear order <, and REF (L) consists of the universal closure of formulas of the form ∃x∀y1 < x · ..."
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This paper develops the model theory of ordered structures that satisfy the analogue, REF (L), of the reflection principle of ZermeloFraenkel set theory. Here L is a language with a distinguished linear order <, and REF (L) consists of the universal closure of formulas of the form ∃x∀y1 < x · · · ∀y1 < x ϕ(y1, · · ·, yn) ↔ ϕ <x (y1, · · ·, yn), where ϕ(y1, · · ·, yn) is an Lformula, ϕ <x is the Lformula obtained by restricting all the quantifiers of ϕ to the initial segment determined by x, and x is a variable that does appear in ϕ. Our results include: Theorem. The following five conditions are equivalent for a complete first order theory T in a countable language L with a distinguished linear order: (1) Some model of T has an elementary end extension with a first new element. (2) T ⊢ REF (L). (3) T has an ω1like model that continuously embeds ω1. (4) For some regular uncountable cardinal κ, T has a κlike model that continuously embeds a stationary subset of κ. (5) For some regular uncountable cardinal κ, T has a κlike model M that has an elementary extension in which the supremum of M exists. Moreover, if κ is a regular cardinal satisfying κ = κ <κ, then each of the above conditions is equivalent to: (6) T has a κ +like model that continuously embeds a stationary subset of κ.