Troublesome Quasi-Cardinals and the Axiom of Choice

Abstract

The article concerns French–Krause quasi-set theory $\mathfrak {Q}$, in particular, its very controversial axioms on quasi-cardinals, the Axiom of Choice and the Weak Extensionality Axiom. A modification $\mathfrak {Q}^{\ast }$ of $\mathfrak {Q}$ is proposed. Similarly to what is going on in $\mathfrak {Q}$, indiscernibility is a primitive concept of $\mathfrak {Q}^{\ast }$, objects of $\mathfrak {Q}^{\ast }$ are either M-atoms or m-atoms, or quasi-classes. Some quasi-classes are quasi-sets. The ZFA-kernel of $\mathfrak {Q}^{\ast }$ is defined to serve as a model of Zermelo–Fraenkel set theory with atoms (usually denoted by ZFA or ZFU). The Axiom of Choice is not an axiom of $\mathfrak {Q}^{\ast }$. For a quasi-set x and a finite ordinal n, qc(x, n) is the statement: “The quasi-set x has its quasi-cardinal equal to n”. Axioms on the statements qc(x, n) are formulated in such a way that, for every n ∈ ω, if x is in the ZFA-kernel of $\mathfrak {Q}^{\ast }$, then it holds in $\mathfrak {Q}^{\ast }$ that qc(x, n) is true if and only if x is a finite set equipotent to n; however, equipotence does not appear in the axioms concerning the statements qc(x, n). Concepts of strongly finite, strongly infinite, strongly countable and strongly uncountable quasi-sets are proposed. A Proper Weak Extensionality Principle is introduced. One of the consequences of this principle is a theorem on the unobservability of permutations.

Dedicated to Décio Krause on his 70th birthday.