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

Classical Set Theory: Theory of Sets and Classes

This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.Comment: 162 page

Solutions of Extension and Limits of Some Cantorian Paradoxes

Cantor thought of the principles of set theory or intuitive principles as universal forms that can apply to any actual or possible totality. This is something, however, which need not be accepted if there are totalities which have a fundamental ontological value and do not conform to these principles. The difficulties involved are not related to ontological problems but with certain peculiar sets, including the set of all sets that are not members of themselves, the set of all sets, and the ordinal of all ordinals. These problematic totalities for intuitive theory can be treated satisfactorily with the Zermelo and Fraenkel (ZF) axioms or the von Neumann, Bernays, and Gödel (NBG) axioms, and the iterative conceptions expressed in them

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page