Abstract

The article concerns French–Krause quasi-set theory????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}$$\end{document}, in particular, its very controversial axioms on quasi-cardinals, the Axiom of Choice and the Weak Extensionality Axiom. A modification????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document} of????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}$$\end{document} is proposed. Similarly to what is going on in????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}$$\end{document}, indiscernibility is a primitive concept of????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document}, objects of????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document} are either M-atoms or m-atoms, or quasi-classes. Some quasi-classes are quasi-sets. The ZFA-kernel of????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document} 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????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document}. 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????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document}, then it holds in????∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}^{\ast }$$\end{document} 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.