11 found
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  1.  14
    Reverse Mathematics and Uniformity in Proofs Without Excluded Middle.Jeffry L. Hirst & Carl Mummert - 2010 - Notre Dame Journal of Formal Logic 52 (2):149-162.
    We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA ${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. (...)
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  2. Reverse Mathematics and Π21 Comprehension.Carl Mummert & Stephen G. Simpson - 2005 - Bulletin of Symbolic Logic 11 (4):526-533.
    We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π2 1 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by $\lbrace N_p \mid p \in P \rbrace$ . Here P is a poset, MF(P) is the set of maximal filters on P, and $N_p = \lbrace F \in MF(P) \mid p \in F \rbrace$ . If the poset P is countable, (...)
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  3.  9
    Reverse Mathematics of Mf Spaces.Carl Mummert - 2006 - Journal of Mathematical Logic 6 (2):203-232.
  4.  19
    Subsystems of Second-Order Arithmetic Between RCA0 and WKL0.Carl Mummert - 2008 - Archive for Mathematical Logic 47 (3):205-210.
    We study the Lindenbaum algebra ${\fancyscript{A}}$ (WKL o, RCA o) of sentences in the language of second-order arithmetic that imply RCA o and are provable from WKL o. We explore the relationship between ${\Sigma^1_1}$ sentences in ${\fancyscript{A}}$ (WKL o, RCA o) and ${\Pi^0_1}$ classes of subsets of ω. By applying a result of Binns and Simpson (Arch. Math. Logic 43(3), 399–414, 2004) about ${\Pi^0_1}$ classes, we give a specific embedding of the free distributive lattice with countably many generators into ${\fancyscript{A}}$ (...)
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  5.  25
    An Incompleteness Theorem for [Image].Carl Mummert & Stephen G. Simpson - 2004 - Journal of Symbolic Logic 69 (2):612 - 616.
    Let n be a positive integer. By a $\beta_{n}-model$ we mean an $\omega-model$ which is elementary with respect to $\sigma_{n}^{1}$ formulas. We prove the following $\beta_{n}-model$ version of $G\ddot{o}del's$ Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a $\beta_{n}-model$ of S, then there exists a $\beta_{n}-model$ of S + "there is no countable $\beta_{n}-model$ of S". We also prove a $\beta_{n}-model$ version of $L\ddot{o}b's$ Theorem. As a corollary, we obtain (...)
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  6.  12
    The Modal Logic of Reverse Mathematics.Carl Mummert, Alaeddine Saadaoui & Sean Sovine - 2015 - Archive for Mathematical Logic 54 (3-4):425-437.
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  7.  9
    Filters on Computable Posets.Steffen Lempp & Carl Mummert - 2006 - Notre Dame Journal of Formal Logic 47 (4):479-485.
    We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle (...)
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  8.  4
    Reverse Mathematics and Properties of Finite Character.Damir D. Dzhafarov & Carl Mummert - 2012 - Annals of Pure and Applied Logic 163 (9):1243-1251.
  9.  1
    An Incompleteness Theorem for Βn-Models.Carl Mummert & Stephen G. Simpson - 2004 - Journal of Symbolic Logic 69 (2):612-616.
    Let n be a positive integer. By a βn-model we mean an ω-model which is elementary with respect to Σ1n formulas. We prove the following βn-model version of Gödel’s Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a βn-model of S, then there exists a βn-model of S + “there is no countable βn-model of S”. We also prove a βn-model version of Löb’s Theorem. As a corollary, we obtain (...)
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  10. An Incompleteness Theorem for Β N -Models.Carl Mummert & Stephen G. Simpson - 2004 - Journal of Symbolic Logic 69 (2):612-616.
  11. Reverse Mathematics And Π 12 Comprehension.Carl Mummert & Stephen Simpson - 2005 - Bulletin of Symbolic Logic 11 (3):526-533.
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