12 research outputs found

    The weakness of the pigeonhole principle under hyperarithmetical reductions

    Full text link
    The infinite pigeonhole principle for 2-partitions (RT21\mathsf{RT}^1_2) asserts the existence, for every set AA, of an infinite subset of AA or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that RT21\mathsf{RT}^1_2 admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every Δn0\Delta^0_n set, of an infinite lown{}_n subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page

    On uniform relationships between combinatorial problems

    Get PDF
    The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by Q, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n; j; k 1, if j < k then Ramsey's theorem for n-tuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for n-tuples and j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely ner metric by which to gauge the relative strength of mathematical propositions. We also study Weak K�onig's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve in nitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of di erence between combinatorial problems previously thought to be more closely related

    Automated Deduction – CADE 28

    Get PDF
    This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

    Categories and logical syntax

    Get PDF
    The notions of category and type are here studied through the lens of logical syntax: Aristotle's as well as Kant's categories through the traditional form of proposition `S is P', and modern doctrines of type through the Fregean form of proposition `F(a)', function applied to argument. Topics covered include the conception of categories as highest genera; the parts of speech and their relation to categories; the attempt to derive categories from more fundamental notions; the notion of a range of significance; the notion of a type assignment; sortal concepts and the notions of identity and generality; and the distinction between formal and material categories.UBL - phd migration 201
    corecore