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

• Autorzy: Paul Howard , Eleftherios Tachtsis
• Języki publikacji: EN

Abstrakty

EN

We investigate, in set theory without the Axiom of Choice 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 𝖹𝖥 (Zermelo-Fraenkel set theory minus 𝖠𝖢), equivalent to the Boolean Prime Ideal Theorem 𝖡𝖯𝖨, whereas
(2) Q(2) is strictly weaker than 𝖡𝖯𝖨 in 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 𝖡𝖯𝖨.

Kategorie tematyczne

• 03E25: Axiom of choice and related propositions
• 54D30: Compactness
• 54B10: Product spaces
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2016
• Tom: 234
• Numer: 3
• Strony: 241-252

Daty

wydano

2016

Twórcy

autor

Paul Howard

• Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U.S.A.

autor

Eleftherios Tachtsis

• Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece

Identyfikatory

DOI

10.4064/fm961-1-2016