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  1. A Simple Proof and Some Difficult Examples for Hindman's Theorem.Henry Towsner - 2012 - Notre Dame Journal of Formal Logic 53 (1):53-65.
    We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several examples of colorings of the integers which do not have computable witnesses to Hindman's Theorem.
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  • Non-Principal Ultrafilters, Program Extraction and Higher-Order Reverse Mathematics.Alexander P. Kreuzer - 2012 - Journal of Mathematical Logic 12 (1):1250002-.
    We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher-order arithmetic. Let [Formula: see text] be the statement that a non-principal ultrafilter on ℕ exists and let [Formula: see text] be the higher-order extension of ACA0. We show that [Formula: see text] is [Formula: see text]-conservative over [Formula: see text] and thus that [Formula: see text] is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly (...)
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  • Ultrafilters in Reverse Mathematics.Henry Towsner - 2014 - Journal of Mathematical Logic 14 (1):1450001.
    We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA0, ATR0, and [Formula: see text].
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