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

100%
Howard P., Tachtsis E.
Fundamenta Mathematicae
|
2016
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tom 234
|
nr 3
241-252
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.
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On rigid relation principles in set theory without the axiom of choice

100%
Howard P., Tachtsis E.
Fundamenta Mathematicae
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2016
|
tom 232
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nr 3
199-226
EN
We study the deductive strength of the following statements: 𝖱𝖱: every set has a rigid binary relation, 𝖧𝖱𝖱: every set has a hereditarily rigid binary relation, 𝖲𝖱𝖱: every set has a strongly rigid binary relation, in set theory without the Axiom of Choice. 𝖱𝖱 was recently formulated by J. D. Hamkins and J. Palumbo, and 𝖲𝖱𝖱 is a classical (non-trivial) 𝖹𝖥𝖢-result by P. Vopěnka, A. Pultr and Z. Hedrlín.
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On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

81%
Herrlich H., Howard P., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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tom 63
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nr 2
89-112
EN
We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice (AC), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y → X.
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