A Single Axiom for Set Theory

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)

Set Theory Without the Axiom of Foundation

Declaration I declare that this essay is work done as part of the Part III Examination. It is the result of my own work, and except where stated otherwise, includes nothing which was performed in collaboration. No part of this essay has been submitted for a degree or any such qualification. Self-reference has been a concept which has concerned philosophers for millennia, and which gives rise to...

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 ...