Conservative extensions of models of set theory and generalizations

1986 ◽  
Vol 51 (4) ◽  
pp. 1005-1021 ◽  
Author(s):  
Ali Enayat
Keyword(s):  
Set Theory ◽  
Satisfying Condition ◽  
Peano Arithmetic ◽  
Negative Answer ◽  
Compactness Theorem ◽  
Elementary Extension ◽  
Direct Contrast ◽  
Conservative Extensions ◽  
Models Of Set Theory ◽  
Completeness Theorems

An attempt to answer the following question gave rise to the results of the present paper. Let be an arbitrary model of set theory. Does there exist an elementary extension of satisfying the two requirements: (1) contains an ordinal exceeding all the ordinals of ; (2) does not enlarge any (hyper) integer of ? Note that a trivial application of the ordinary compactness theorem produces a model satisfying condition (1); and an internal ultrapower modulo an internal ultrafilter produces a model satisfying condition (2) (but not (1), because of the axiom of replacement). Also, such a satisfying both conditions (1) and (2) exists if the external cofinality of the ordinals of is countable, since by [KM], would then have an elementary end extension.Using a class of models constructed by M. Rubin using in [RS], and already employed in [E1], we prove that our question in general has a negative answer (see Theorem 2.3). This result generalizes the results of M. Kaufmann and the author (appearing respectively in [Ka] and [E1]) concerning models of set theory with no elementary end extensions.In the course of the proof it was necessary to establish that all conservative extensions (see Definition 2.1) of models of ZF must be cofinal. This is in direct contrast with the case of Peano arithmetic where all conservative extensions are end extensional (as observed by Phillips in [Ph1]). This led the author to introduce two useful weakenings of the notion of a conservative end extension which, as shown by the “completeness” theorems in §3, can exist.

scholarly journals Chains of end elementary extensions of models of set theory

1998 ◽  
Vol 63 (3) ◽  
pp. 1116-1136 ◽  
Author(s):  
Andrés Villaveces
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Elementary Extension ◽  
Models Of Set Theory

AbstractLarge cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (‘unfoldable cardinals’) lie in the boundary of the propositions consistent with ‘V = L’ and the existence of 0#. We also provide an ‘embedding characterisation’ of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

A Boolean ultrapower which is not an ultrapower

1976 ◽  
Vol 41 (1) ◽  
pp. 245-249 ◽  
Author(s):  
Bernd Koppelberg ◽  
Sabine Koppelberg
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Natural Generalization ◽  
Negative Answer ◽  
Complete Boolean Algebra ◽  
Image Position ◽  
Models Of Set Theory

Several people have independently been studying Boolean ultrapowers recently; see for example [2], [3], [4], [6]. Boolean ultrapowers are a quite natural generalization of the well-known usual ultrapowers, but it seemed to be unknown whether every Boolean ultrapower is isomorphic to an ultrapower. We give a negative answer to that question. We further show that a Boolean ultrapower by an ℵ1-regular ultrafilter need not be ℵ2-universal, i.e. that Theorem 4.3.12 of [1] does not hold for Boolean ultrapowers.Let B be a complete Boolean algebra (we identify the algebra with its underlying set), whose operations are denoted by +, ·, −, 0, 1, Σ, Π Let be a structure for some language ℒ. For those who are familiar with Boolean-valued models of set theory, the B-valued model may be described by its underlying setandif R is an n-place relation in ℒ or equality, its interpretation in , u1 … un Є M(B).

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 Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

On external Scott algebras in nonstandard models of Peano arithmetic

1996 ◽  
Vol 61 (2) ◽  
pp. 586-607
Author(s):  
Vladimir Kanovei
Keyword(s):  
Peano Arithmetic ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Nonstandard Models ◽  
Necessary And Sufficient

AbstractWe prove that a necessary and sufficient condition for a countable set of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: is closed under arithmetical definability and contains 0(ω) the set of all (Gödel numbers of) true arithmetical sentences.Some results related to definability of sets of integers in elementary extensions of ω are included.

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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