ZF and the axiom of choice in some paraconsistent set theories

Zf and the Axiom of Choice in Some Paraconsistent Set Theories

In this paper, we present set theories based upon the paraconsistent logic Pac. We describe two different techniques to construct models of such set theories. The first of these is an adaptation of one used to construct classical models of positive comprehension. The properties of the models obtained in that way give rise to a natural paraconsistent set theory which is presented here. The statu...

The axiom of choice and the law of excluded middle in weak set theories

In constructive mathematics the axiom of choice (AC) has a somewhat ambiguous status. On the one hand, in intuitionistic set theory, or the local set theory associated with a topos ([2]) it can be shown to entail the law of excluded middle (LEM) ([ 3 ], [ 5 ]). On the other hand, under the “propositions-as types” interpretation which lies at the heart of constructive predicative type theories s...

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)

Paradox, ZF, and the Axiom of Foundation ∗

The Relative Consistency of the Axiom of Choice - Mechanized Using Isabelle/ZF

The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem [15]. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems with...