In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.

This article proposes a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor’s account if the axiom of choice is true, but its consequences differ from those of Cantor’s if the axiom of choice is false. This article argues that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set (...) that is bigger than it—that can arise from Cantor’s account if the axiom of choice is false, illuminates the debate over whether the axiom of choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

We show that the both assertions “in every vector space B over a finite element field every subspace V ⊆ B has a complementary subspace S” and “for every family [MATHEMATICAL SCRIPT CAPITAL A] of disjoint odd sized sets there exists a subfamily ℱ={Fj:j ϵω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ℚ every generating set includes a basis”.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible principles. (...) Some of them concern the interaction between propositions and worlds, and they appear to afford the means to map classes of propositions into propositions – or, likewise, classes of worlds into worlds – in a one-to-one fashion that leads to contradiction. Yet another family of paradoxes threaten the view that whatever could exist does, in fact, exist, which is in line with modal realism, for example. This article aims to survey and identify the source of each family of paradoxes as well as to outline some responses to them. (shrink)

We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.

We show that the Axiom of Choice is equivalent to each of the following statements: A product of closures of subsets of topological spaces is equal to the closure of their product ; A product of complete uniform spaces is complete.

Frege introduced the notion of pragmatic force as what distinguishes statements from questions. This distinction was elaborated by Wittgenstein in his later works, and systematised as an account of different kinds of speech acts in formal dialogue theory by Hamblin. It lies at the heart of the inferential semantics more recently developed by Brandom. The present paper attempts to sketch some of the relations between these developments.

We try to recast in modern terms a choice principle conceived by Beppo Levi, who called it the Approximation Principle. Up to now, there was almost no discussion about Levi’s contribution, due to the quite obscure formulation of AP the author has chosen. After briefly reviewing the historical and philosophical surroundings of Levi’s proposal, we undertake our own attempt at interpreting AP. The idea underlying the principle, as well as the supposed faithfulness of our version to Levi’s original intention, are (...) then discussed. Finally, an application of AP to a property of metric spaces is presented, with the aim of showing how AP may work in contexts where other forms of choice are commonly at use. (shrink)