Book ChapterDOI
Infinitary Methods in the Model Theory of Set Theory
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In this article, the authors discuss the infinitary methods in the model theory of set theory and also present the study of the end extensions of models of Zermelo-Fraenkel (ZF) set theory.Abstract:
Publisher Summary This chapter discusses the infinitary methods in the model theory of set theory and also presents the study of the end extensions of models of Zermelo-Fraenkel (ZF) set theory. The theorem of Keisler-Morley, which states that every countable model of ZF has a proper elementary end extension, is focussed. It is shown that if ZF is consistent then there are uncountable models of ZF with no end extensions. The necessary preliminaries are described. All the results are proved using methods and results from infinitary logic. Some of the result on collapsing cardinals used the compactness theorem to prove the theorem of Friedman.read more
Citations
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Journal ArticleDOI
Scott sentences and admissible sets
TL;DR: In this article, bounds are obtained for the Scott sentence of a structure and special attention is given to linear orderings, where the existence of Scott sentences is related to other properties admissible sets.
Journal ArticleDOI
Partially conservative extensions of arithmetic
TL;DR: In this paper, it was shown that any sentence X in r which is provably equivalent to a sentence in any class "simpler" than r is not conservative for classes "more complicated" than I, unless r and r' are so related that such a 0 cannot exist.
Book ChapterDOI
Choice Functions on Sets and Classes
TL;DR: The axiomatic set theory of Zermelo-Fraenkel (ZF) is based on the idea of limitation of size as discussed by the authors, and a list of principles for set formation is given such as union, power set, and replacement; and the universe V of all sets is built up in stages V. The aim of the chapter is to discuss these axioms of choice and related versions of choice, and investigate their relative strength.
Book ChapterDOI
On extendability of models of ZF set theory to the models of Kelley-Morse theory of classes
W. Marek,A. Mostowski +1 more
TL;DR: In this paper, the authors deal with the four basic theories: ZFC (theory of Zermelo and Fraenkel with choice), ZF KM, KM (Kelley-Morse), and KMC (KMC).
References
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Journal ArticleDOI
Infinitary logic and admissible sets
TL;DR: In recent years, much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways as mentioned in this paper, motivated by the desire to find a language which is (I) strong enough to express interesting properties not expressible by the classical language, but (II) still simple enough to yield interesting general results.
Journal ArticleDOI
Elementary extensions of models of set theory
H. Jerome Keisler,Michael Morley +1 more
TL;DR: Model Theoretic methods are used to extend models of set theory while leaving specified sets fixed as mentioned in this paper, where every countable modelU of ZF has an extension leaving every set in U fixed, and for each (inU) regular cardinala an extension enlarginga but leaving each cardinal less than a fixed.