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Partial choice functions for families of finite sets

100%

Hall E., Shelah S.

Fundamenta Mathematicae

2013

tom 220

nr 3

207-216

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 space over the finite field $𝔽_{p}$. The use of atoms is then eliminated by citing an embedding theorem of Pincus. In the case where m is not prime, suitable permutation models are built from the models used in the prime cases.

Definitions of finiteness based on order properties

81%

De la Cruz O., Dzhafarov D., Hall E.

Fundamenta Mathematicae

2006

tom 189

nr 2

155-172

A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them under subsets or under quotients. We work in set theory without AC to establish relations of implication and independence between these definitions, as well as between them and other notions of finiteness previously studied in the literature. It turns out that several well known definitions of finiteness (including Dedekind finiteness) fit into our framework by being equivalent to one of our definitions; however, a few of our definitions are actually new. We also show that Ia-finite unions of Ia-finite sets are P-finite (one of our new definitions), but that the class of P-finite sets is not provably closed under unions.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Hall E.

1 De la Cruz O.

1 Dzhafarov D.

1 Shelah S.

1 2013

1 2006

Wyniki wyszukiwania

Partial choice functions for families of finite sets

Definitions of finiteness based on order properties