9,925 research outputs found

    On Rules and Parameter Free Systems in Bounded Arithmetic

    Get PDF
    We present model–theoretic techniques to obtain conservation results for first order bounded arithmetic theories, based on a hierarchical version of the well known notion of an existentially closed model.Ministerio de Educación y Ciencia MTM2005-0865

    Existentially Closed Models and Conservation Results in Bounded Arithmetic

    Get PDF
    We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.Ministerio de Educación y Ciencia MTM2005-08658Junta de Andalucía TIC-13

    Reverse mathematics and properties of finite character

    Get PDF
    We study the reverse mathematics of the principle stating that, for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey's lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.Comment: This paper corresponds to section 4 of arXiv:1009.3242, "Reverse mathematics and equivalents of the axiom of choice", which has been abbreviated and divided into two pieces for publicatio

    Definable orthogonality classes in accessible categories are small

    Get PDF
    We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This result follows from the fact that cohomology equivalences are \Sigma_2. In contrast with this fact, homology equivalences are \Sigma_1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies have been correcte
    • …
    corecore