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Partial choice functions for families of finite sets

Eric J. Hall, Saharon Shelah (2013)

Fundamenta Mathematicae

Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector...

Perfect set properties in models of ZF

Carlos Augusto Di Prisco, Franklin C. Galindo (2010)

Fundamenta Mathematicae

We study several perfect set properties of the Baire space which follow from the Ramsey property ω → ( ω ) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

Prime Ideal Theorems and systems of finite character

Marcel Erné (1997)

Commentationes Mathematicae Universitatis Carolinae

We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if S is a system of finite character then so is the system of all collections of finite subsets of ⋃ S meeting a common member of S ), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these statements...

Products of Lindelöf T 2 -spaces are Lindelöf – in some models of ZF

Horst Herrlich (2002)

Commentationes Mathematicae Universitatis Carolinae

The stability of the Lindelöf property under the formation of products and of sums is investigated in ZF (= Zermelo-Fraenkel set theory without AC, the axiom of choice). It is • not surprising that countable summability of the Lindelöf property requires some weak choice principle, • highly surprising, however, that productivity of the Lindelöf property is guaranteed by a drastic failure of AC, • amusing that finite summability of the Lindelöf property takes place if either some weak choice principle...

Products, the Baire category theorem, and the axiom of dependent choice

Horst Herrlich, Kyriakos Keremedis (1999)

Commentationes Mathematicae Universitatis Carolinae

In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (i) The axiom of dependent choice. (ii) Products of compact Hausdorff spaces are Baire. (iii) Products of pseudocompact spaces are Baire. (iv) Products of countably compact, regular spaces are Baire. (v) Products of regular-closed spaces are Baire. (vi) Products of Čech-complete spaces are Baire. (vii) Products of pseudo-complete spaces are Baire.

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