Riesz Theory Without Axiom of Choice

Rowbottom cardinals without the Axiom of Choice

We show that for all natural numbers n, the theory “ZF +DCאn + אω is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + אω1 is an ω2-Rowbottom cardinal carrying an ω2-Rowbottom filter and ω1 is regular” has the same consistency strength as the theory “ZFC + There...

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...

Expected utility theory without the completeness axiom

Universidad de Montevideo and New York University Istituto di Metodi Quantitativi Università Bocconi Department of Economics New York University September, 2001 for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-...

The Axiom of Choice from the Viewpoint of Recursion Theory