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L’« axiome du choix simple » est le principe selon lequel on peut choisir un élément dans tout ensemble non vide. Cet « autre axiome du choix » a une histoire paradoxale et riche, dont la première partie de cet article recherche les traces et repère les enjeux. Apparaissent comme décisifs le statut de la théorie des ensembles dans les mathématiques intuitionnistes, mais aussi la tension croissante entre technicisation de la logique et réflexion épistémologique des mathématiciens. La deuxième partie...
In set theory without the Axiom of Choice ZF, we prove that for every commutative field , the following statement : “On every non null -vector space, there exists a non null linear form” implies the existence of a “-linear extender” on every vector subspace of a -vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically...
We work in set-theory without choice ZF. Given a commutative field , we consider the statement : “On every non null -vector space there exists a non-null linear form.” We investigate various statements which are equivalent to in ZF. Denoting by the two-element field, we deduce that implies the axiom of choice for pairs. We also deduce that implies the axiom of choice for linearly ordered sets isomorphic with .
In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles.
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.
(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of to Y has size ≤ |(ω)|”.
(ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”.
(iii) Under P(ω) the following two statements are equivalent:
(a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter.
(b) Every Boolean algebra of...
In set theory without the axiom of choice (), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC (AC for linearly ordered families of nonempty sets)—and hence AC (AC for well-ordered families of nonempty sets)— (where is an uncountable regular cardinal), and “for every infinite set , there is a bijection ”, implies the statement “there exists a field such that every vector...
We study in ZF and in the class of spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.
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.
Let I ⊆ P(ω) be an ideal. We continue our investigation of the class of spaces with the I-ideal convergence property, denoted (I). We show that if I is an analytic, non-countably generated P-ideal then (I) ⊆ s₀. If in addition I is non-pathological and not isomorphic to , then (I) spaces have measure zero. We also present a characterization of the (I) spaces using clopen covers.
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 has a ternary partition (see Section 1, Definition 2) then the Russell cardinal 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...
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space is countably compact if and only if it is countably subcompact relative to . (iii) For every metrizable space , the following are equivalent:
(a) is compact;
(b) for every open filter of , ;
(c) is subcompact relative to . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " is countably compact" and " is compact"
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