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Jech's research also includes mathematical logic, algebra, analysis, topology, and measure theory. Jech gave the first published proof of the consistency of the existence of a Suslin line. With Karel Prikry, he introduced the notion of precipitous ideal. He gave several models where the axiom of choice failed, for example one with ω 1 measurable.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
1999: (with Thomas Jech) Introduction to Set Theory, Third edition. Monographs and Textbooks in Pure and Applied Mathematics, 220. Marcel Dekker ISBN 0-8247-7915-0 [3] [4] [5] 1992: (with David Ballard) "Standard foundations for nonstandard analysis", Journal of Symbolic Logic 57(2): 741–748 MR 1169206
Pocket set theory; Positive set theory; S (Boolos 1989) Scott–Potter set theory; Tarski–Grothendieck set theory; Von Neumann–Bernays–Gödel set theory; Zermelo–Fraenkel set theory; Zermelo set theory; Set (mathematics) Set-builder notation; Set-theoretic topology; Simple theorems in the algebra of sets; Subset; Θ (set theory) Tree ...
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
In mathematics, forcing is a method of constructing new models M[G] of set theory by adding a generic subset G of a poset P to a model M.The poset P used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable P.
Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. Mark Howard, Applications of Fodor's Lemma to Vaught's Conjecture. Ann. Pure and Appl. Logic 42(1): 1-19 (1989). Simon Thomas, The Automorphism Tower Problem. PostScript file at ; S. Todorcevic, Combinatorial dichotomies in set theory. pdf at
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. [3] Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language.
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