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

Paul HowardEleftherios Tachtsis — 2016

Fundamenta Mathematicae

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

On rigid relation principles in set theory without the axiom of choice

Paul HowardEleftherios Tachtsis — 2016

Fundamenta Mathematicae

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.

On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

Horst HerrlichPaul HowardEleftherios Tachtsis — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

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.

On special partitions of Dedekind- and Russell-sets

Horst HerrlichPaul HowardEleftherios Tachtsis — 2012

Commentationes Mathematicae Universitatis Carolinae

A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal a has a ternary partition (see Section 1, Definition 2) then the Russell cardinal a + 2 fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell...

On preimages of ultrafilters in ZF

Horst HerrlichPaul HowardKyriakos Keremedis — 2016

Commentationes Mathematicae Universitatis Carolinae

We show that given infinite sets X , Y and a function f : X Y which is onto and n -to-one for some n , the preimage of any ultrafilter of Y under f extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms A and a finite-to-one onto function f : A ω such that for each free ultrafilter of ω its preimage under f does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact pseudometric...

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