Axiomatic Set Theories.

Import 06/11/2014Práce je zaměřena na axiomatické teorie množin. Jsou v ní přehledně zpracovány a popsány nejznámější teorie jako Zermelo-Fraenkelova teorie množin, Gödel-Bernaysova teorie množin a Kelley-Morseova teorie množin. V úvodu je popsána výstavba logické teorie s definicí základních pojmů a principů. Dále je zde zhrnut a objasněn důvod vzniku axiomatických teorií množin a důvod, proč naivní teorie množin nebyla dostačující. Další kapitoly jsou zaměřeny na důkazy jednotlivých teorémů u konkrétních teorií množin. Jedna z těchto kapitol je zaměřena na zdůvodnění, proč můžeme tvrdit, že Kelley-Morseova teorie množin je silnější než Zermelo-Fraenkelova teorie množin. Další kapitola popisuje, jakým způsobem omezit Gödel-Bernaysovu teorii množin na teorii s konečným počtem axiomů, i když se tato teorie standardně považuje za nekonečně axiomatizovatelnou. Poslední kapitola je věnována důležitým důkazům bezespornosti axiomu výběru a hypotézy kontinua se Zermelo-Fraenkelovou teorií množin.Thesis focuses on axiomatic set theories. There are clearly described and processed the most famous theories - Zermelo-Fraenkel set theory, Gödel-Bernays set theory and Kelley-Morse set theory. In introduction is described the construction of logical theory with the definition of basic concepts and principles. There is also summarized and explained the cause of formation the axiomatic set theory and why naive set theory was not sufficient. Next chapters focus on proofs of individual theorems in specific theories. One of these chapters focuses on the rationale, why we can say that the Kelley-Morse set theory is stronger than Zermelo-Fraenkel set theory. Another chapter describes, how is possible to limit Gödel-Bernays set theory to theory with a finite number of axioms, although this theory is normally considered as infinitely axiomatized. The last chapter is dedicated to important proofs of unequivocalness of axiom of choice and the continuum hypothesis with Zermelo-Fraenkel set theory.460 - Katedra informatikyvelmi dobř

Proof theory of weak compactness

We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations

The Essence of Intuitive Set Theory

Intuitive Set Theory (IST) is defined as the theory we get, when we add Axiom of Monotonicity and Axiom of Fusion to Zermelo-Fraenkel set theory. In IST, Continuum Hypothesis is a theorem, Axiom of Choice is a theorem, Skolem paradox does not appear, nonLebesgue measurable sets are not possible, and the unit interval splits into a set of infinitesimals

Derived rules for predicative set theory: an application of sheaves

We show how one may establish proof-theoretic results for constructive Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and the Bar Induction rule for Baire space, by constructing sheaf models and using their preservation properties

Applying SMT Solvers to the Test Template Framework

The Test Template Framework (TTF) is a model-based testing method for the Z notation. In the TTF, test cases are generated from test specifications, which are predicates written in Z. In turn, the Z notation is based on first-order logic with equality and Zermelo-Fraenkel set theory. In this way, a test case is a witness satisfying a formula in that theory. Satisfiability Modulo Theory (SMT) solvers are software tools that decide the satisfiability of arbitrary formulas in a large number of built-in logical theories and their combination. In this paper, we present the first results of applying two SMT solvers, Yices and CVC3, as the engines to find test cases from TTF's test specifications. In doing so, shallow embeddings of a significant portion of the Z notation into the input languages of Yices and CVC3 are provided, given that they do not directly support Zermelo-Fraenkel set theory as defined in Z. Finally, the results of applying these embeddings to a number of test specifications of eight cases studies are analysed.Comment: In Proceedings MBT 2012, arXiv:1202.582