On the set-theoretic strength of the $n$-compactness of generalized Cantor cubes

Paul Howard¹ ; Eleftherios Tachtsis²

¹ Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197, U.S.A.
² Department of Mathematics University of the Aegean Karlovassi 83200, Samos, Greece

Fundamenta Mathematicae, Tome 234 (2016) no. 3, pp. 241-252

Résumé

We investigate, in set theory without the Axiom of Choice $\mathbf{\mathsf{AC}}$, the set-theoretic strength of the statement $Q(n)$: For every infinite set $X$, the Tychonoff product $2^X$, where $2=\{0,1\}$ has the discrete topology, is $n$-compact, where $n=2,3,4,5$ (definitions are given in Section 1). We establish the following results: (1) For $n=3,4,5$, $Q(n)$ is, in $\mathbf{\mathsf{ZF}}$ (Zermelo–Fraenkel set theory minus $\mathbf{\mathsf{AC}}$), equivalent to the Boolean Prime Ideal Theorem $\mathbf{\mathsf{BPI}}$, whereas (2) $Q(2)$ is strictly weaker than $\mathbf{\mathsf{BPI}}$ in $\mathbf{\mathsf{ZFA}}$ set theory (Zermelo–Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms). This settles the open problem in Tachtsis (2012) on the relation of $Q(n)$, $n=2,3,4,5$, to $\mathbf{\mathsf{BPI}}$.

Détail

DOI : 10.4064/fm961-1-2016

Keywords: investigate set theory without axiom choice mathbf mathsf set theoretic strength statement every infinite set tychonoff product where has discrete topology n compact where definitions given section establish following results mathbf mathsf zermelo fraenkel set theory minus mathbf mathsf equivalent boolean prime ideal theorem mathbf mathsf bpi whereas strictly weaker mathbf mathsf bpi mathbf mathsf zfa set theory zermelo fraenkel set theory axiom extensionality weakened order allow atoms settles problem tachtsis relation nbsp mathbf mathsf bpi