Michael O. Rabin. On recursively enumerable and arithmetic models of set theory. The Journal of symbolic logic, vol. 23 no. 4 (for 1958, pub. 1959), pp. 408–416.

1963 ◽  
Vol 28 (2) ◽  
pp. 167-168
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Symbolic Logic ◽  
Recursively Enumerable ◽  
Models Of Set Theory

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.

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

Subdifferentials in Boolean-valued models of set theory

1984 ◽  
Vol 24 (5) ◽  
pp. 735-746 ◽  
Author(s):  
A. G. Kusraev ◽  
S. S. Kutateladze
Keyword(s):  
Set Theory ◽  
Models Of Set Theory
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