3,048 research outputs found

    Reverse mathematics and well-ordering principles

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    The paper is concerned with generally Pi^1_2 sentences of the form 'if X is well ordered then f(X) is well ordered', where f is a standard proof theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded omega-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we shall focus on the well-known psi-function which figures prominently in so-called predicative proof theory. However, the approach taken here lends itself to generalization in that the techniques we employ can be applied to many other proof-theoretic functions associated with cut elimination theorems. In this paper we show that the statement 'if X is well ordered then 'X0 is well ordered' is equivalent to ATR0. This was first proved by Friedman, Montalban and Weiermann [7] using recursion-theoretic and combinatorial methods. The proof given here is proof-theoretic, the main techniques being Schuette's method of proof search (deduction chains) [13], generalized to omega logic, and cut elimination for infinitary ramified analysis

    Computational reverse mathematics and foundational analysis

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    Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only ω\omega-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a Π11\Pi^1_1 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than Π11-CA0\Pi^1_1\text{-}\mathsf{CA}_0.Comment: Submitted. 41 page

    Reverse mathematics and equivalents of the axiom of choice

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    We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a ⊆\subseteq-maximal subfamily with the finite intersection property and the principle asserting that if PP is a property of finite character then every set has a ⊆\subseteq-maximal subset of which PP holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to Z2\mathsf{Z}_2 to being weaker than ACA0\mathsf{ACA}_0 and incomparable with WKL0\mathsf{WKL}_0. In particular, we identify a choice principle that, modulo Σ20\Sigma^0_2 induction, lies strictly below the atomic model theorem principle AMT\mathsf{AMT} and implies the omitting partial types principle OPT\mathsf{OPT}

    Controlling iterated jumps of solutions to combinatorial problems

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    Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their iterated jumps. For this, we define forcing relations which, unlike Mathias forcing, have the same definitional complexity as the formulas they force. This analysis enables us to answer two questions of Wei Wang, namely, whether cohesiveness and the Erdos-Moser theorem admit preservation of the arithmetic hierarchy, and can be seen as a step towards the resolution of the strictness of the Ramsey-type hierarchies.Comment: 32 page

    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
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