CONTRADICTIONS WITHIN ZERMELO–FRAENKEL SET THEORY WITH AXIOM OF CHOICE

Axiom of Choice in nonstandard set theory

We verify that the best-known nonstandard set theories: IST, BST, and HST, with the Axiom of Choice deleted, are conservative extensions of ZF + Boolean Prime Ideal Theorem. 2010 Mathematics Subject Classification 26E35 (primary); 03E25, 03E70, 03H05 (secondary)

Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice

Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, e...

Mechanising Set Theory: Cardinal Arithmetic and the Axiom of Choice

Formalised Set Theory: Well-Orderings and the Axiom of Choice

In this thesis, we give a substantial formalisation of classical set theory in the proof system Coq. We assume an axiomatisation of ZF and present a development of the theory containing relations, functions and ordinals. The implementation follows the structure of standard text books. In the context of this theory, we prove Zermelo’s Well-Ordering Theorem and the Axiom of Choice equivalent. In ...

Generic Families and Models of Set Theory with the Axiom of Choice

Let M be a countable transitive model of ZFC and i be a countable M -generic family of Cohen reals. We prove that there is no smallest transitive model A' of ZFC that either M u A ç N or A/U {A} ç N . h is also proved that there is no smallest transitive model N of ZFC~ (ZFC theory without the power set axiom) such that M U {A} ç N . It is also proved that certain classes of extensions of M obt...