On recursively enumerable and arithmetic models of set theory

1958 ◽  
Vol 23 (4) ◽  
pp. 408-416 ◽  
Author(s):  
Michael O. Rabin
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Simple Proof ◽  
Side Product ◽  
Logical Constant ◽  
Logical Constants ◽  
Recursive Models ◽  
Recursively Enumerable ◽  
Models Of Set Theory ◽  
Countable Models

In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models.Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.

Completeness properties of heyting's predicate calculus with respect to re models

1976 ◽  
Vol 41 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Simple Proof ◽  
Predicate Calculus ◽  
Recursive Model ◽  
Recursive Models ◽  
Recursively Enumerable ◽  
Heyting Arithmetic ◽  
Recursive Structures ◽  
Classical Predicate

Validity in recursive structures was investigated by several authors. Kreisel [10] has shown that there exists a consistent sentence of classical predicate calculus (CPC) that does not possess a recursive model. The sentence is a conjunction of the axioms of a variant of Bernays set theory, including the axiom of infinity. The language contains additional constants besides ϵ. Later Kreisel [2] and Mostowski [3] presented a sentence (not possessing recursive models) which was a conjunction of axioms of a variant of Bernays set theory without the axiom of infinity but still with additional constants besides ϵ. Later Mostowski [4] improved the result by giving a sentence which can be demonstrated in Heyting arithmetic to be consistent and to have no recursive models. Rabin [6] obtained a simple proof that some sentence of set theory with the single nonlogical constant ϵ does not have any recursively enumerable models.More generally, Mostowski [5] has shown that the set of all sentences valid in all RE models is not arithmetical and Vaught [1] improved this result by showing that it holds for a language of one binary relation. In fact, Vaught gives· a way of translating n-place relations to 2-place ones that preserves the RE characteristic of the model. For further results pertaining to recursive models see Vaught [1].

Elementary extensions of countable models of set theory

1976 ◽  
Vol 41 (1) ◽  
pp. 139-145 ◽  
Author(s):  
John E. Hutchinson
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Countable Model ◽  
Elementary Extension ◽  
Models Of Set Theory ◽  
Countable Models

AbstractWe prove the following extension of a result of Keisler and Morley. Suppose is a countable model of ZFC and c is an uncountable regular cardinal in . Then there exists an elementary extension of which fixes all ordinals below c, enlarges c, and either (i) contains or (ii) does not contain a least new ordinal.Related results are discussed.

scholarly journals EVERY COUNTABLE MODEL OF SET THEORY EMBEDS INTO ITS OWN CONSTRUCTIBLE UNIVERSE

2013 ◽  
Vol 13 (02) ◽  
pp. 1350006 ◽  
Author(s):  
JOEL DAVID HAMKINS
Keyword(s):  
Set Theory ◽  
Order Type ◽  
Large Cardinals ◽  
Countable Model ◽  
Classical Theorem ◽  
Binary Relations ◽  
Finite Sets ◽  
Nonstandard Model ◽  
Models Of Set Theory ◽  
Countable Models

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1 + 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most Ord M; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HF M, ∈M〉 of M. Indeed, 〈 HF M, ∈M〉 is universal for all countable acyclic binary relations.

scholarly journals The classification of countable models of set theory

2020 ◽  
Vol 66 (2) ◽  
pp. 182-189
Author(s):  
John Clemens ◽  
Samuel Coskey ◽  
Samuel Dworetzky
Keyword(s):  
Set Theory ◽  
Models Of Set Theory ◽  
Countable Models

scholarly journals Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

Elementary extensions of models of set theory

2000 ◽  
Vol 39 (7) ◽  
pp. 509-514 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Set Theory ◽  
Models Of Set Theory
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