500 research outputs found

    Representing Scott sets in algebraic settings

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    We prove that for every Scott set SS there are SS-saturated real closed fields and models of Presburger arithmetic

    On structures in hypergraphs of models of a theory

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    We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types of models of a theory, are given

    A saturation property of structures obtained by forcing with a compact family of random variables

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    A method how to construct Boolean-valued models of some fragments of arithmetic was developed in Krajicek (2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random variables defined on a pseudo-finite sample space. We show that under a fairly natural condition on the family (called compactness in K.(2011)) the resulting structure has a property that is naturally interpreted as saturation for existential types. We also give an example showing that this cannot be extended to universal types.Comment: preprint February 201

    Categorical characterizations of the natural numbers require primitive recursion

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    Simpson and the second author asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory RCA0^*_0. We answer in the negative, showing that for any characterization of the natural numbers which is provably true in WKL0^*_0, the categoricity theorem implies Σ10\Sigma^0_1 induction. On the other hand, we show that RCA0^*_0 does make it possible to characterize the natural numbers categorically by means of a set of second-order sentences. We also show that a certain Π21\Pi^1_2-conservative extension of RCA0^*_0 admits a provably categorical single-sentence characterization of the naturals, but each such characterization has to be inconsistent with WKL0^*_0+superexp.Comment: 17 page
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