54 research outputs found

    Unprovability results involving braids

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    We construct long sequences of braids that are descending with respect to the standard order of braids (``Dehornoy order''), and we deduce that, contrary to all usual algebraic properties of braids, certain simple combinatorial statements involving the braid order are true, but not provable in the subsystems ISigma1 or ISigma2 of the standard Peano system.Comment: 32 page

    Connecting the two worlds: well-partial-orders and ordinal notation systems

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    Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc. The maximal order type of a well-partial-order characterizes that order’s strength. Moreover, in many natural cases, a well-partial-order’s maximal order type can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent examples are Friedman’s well-partial-orders of trees with the gap-embeddability relation [76]. The main goal of this dissertation is to investigate a conjecture of Weiermann [86], thereby addressing the problem of the unknown maximal order types and corresponding ordinal notation systems for Friedman’s well-partial orders [76]. Weiermann’s conjecture concerns a class of structures, a typical member of which is denoted by T (W ), each are ordered by a certain gapembeddability relation. The conjecture indicates a possible approach towards determining the maximal order types of the structures T (W ). Specifically, Weiermann conjectures that the collapsing functions #i correspond to maximal linear extensions of these well-partial-orders T (W ), hence also that these collapsing functions correspond to maximal linear extensions of Friedman’s famous well-partial-orders

    Set-Theoretic Geology

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    A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at http://jdh.hamkins.org/set-theoreticgeology

    Cardinals as Ultrapowers : A Canonical Measure Analysis under the Axiom of Determinacy

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    This thesis is in the field of Descriptive Set Theory and examines some consequences of the Axiom of Determinacy concerning partition properties that define large cardinals. The Axiom of Determinacy (AD) is a game-theoretic statement expressing that all infinite two-player perfect information games with a countable set of possible moves are determined, i.e., admit a winning strategy for one of the players. By the term "measure analysis'' we understand the following procedure: given a strong partition cardinal Îș and some cardinal λ > Îș, we assign a measure ” on Îș to λ such that the ultrapower with respect to ” equals λ . A canonical measure analysis is a measure assignment for cardinals larger than a strong partition cardinal Îș and a binary operation on the measures of this assignment that corresponds to ordinal addition on indices of the cardinals. This thesis provides a canonical measure analysis up to the (ω^ω)th cardinal after an odd projective cardinal. Using this canonical measure analysis we show that all cardinals that are ultrapowers with respect to basic order measures are JĂłnsson cardinals. With the canonicity results of this thesis we can state that, if Îș is an odd projective ordinal, Îș^(n), Îș^(ωn+1), and Îș^(ω^n+1), for n<ω, are JĂłnsson under AD

    Pursuit of Wisdom and Quantum Ontology

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    In his late work (De venatione sapientiae), Cusanus unfolded basic ideas of his brilliant theology. After a long period, this ingenious teaching became clearly recognizable especially in our time. Forward with his face to the back, modern scientific theory adopts nowadays a course to which Cusanus had already pointed centuries ago. Modern thought revolves with unexpected precision and unexpected mysteriousness around two issues of his doctrine of wisdom: (i) The possibility-of-being-made is not a figment of the human brain by which it organizes one's thoughts, but a fundamental and indispensable manifestation of reality. (ii) The possibility-of-being-made refers to something antecedent by which both the feasibility and the being-made get their common shape. This ultimate ground embodies the omnipotent oneness in the form of an infinite fund in which the cause of all reality and of all possibility is timelessly stored. Comparisons with the quantum ontology and the theory of quantum gravity impose themselves.Comment: a quantum view to the old teaching of Cusanu
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