92 research outputs found

    Interval orders and reverse mathematics

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    We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2⊕22 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it contains neither 2⊕22 \oplus 2 nor 3⊕13 \oplus 1.Comment: 21 pages; to appear in Notre Dame Journal of Formal Logic; minor changes from the previous versio

    The Veblen functions for computability theorists

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    We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where alpha is a fixed computable ordinal and phi the two-placed Veblen function. For the former statement, we show that omega iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA_0^+ over RCA_0. To prove the latter statement we need to use omega^alpha iterations of the Turing jump, and we show that the statement is equivalent to Pi^0_{omega^alpha}-CA_0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if X is a well-ordering, then so is phi(X,0)" is equivalent to ATR_0 over RCA_0.Comment: 26 pages, 3 figures, to appear in Journal of Symbolic Logi

    Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

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    We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is Π20\Pi^0_2-complete and that the set of Cauchy problems which locally have a unique solution is Σ30\Sigma^0_3-complete. We prove that the set of Cauchy problems which have a global solution is Σ40\Sigma^0_4-complete and that the set of ordinary differential equation which have a global solution for every initial condition is Π30\Pi^0_3-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is Π20\Pi^0_2-complete

    Epimorphisms between linear orders

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    We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.Comment: 15 pages; in version 2 we corrected some typos and rewrote the paragraphs introducing the results of subsection 3.3 (statements and proofs are unchanged

    Borel quasi-orderings in subsystems of second-order arithmetic

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    AbstractWe study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques employed will be the reflection principles and Gandy forcing

    Invariantly universal analytic quasi-orders

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    We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.Comment: 31 pages, 1 figure, to appear in Transactions of the American Mathematical Societ
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