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On the failure of BD-ℕ and BD, and an application to the anti-specker property

  • Robert S. Lubarsky (a1)

Abstract

We give the natural topological model for ¬BD-ℕ, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-ℕ. Also, the natural topological model for ¬BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-ℕ, it is brought out in detail how BD-ℕ fails.

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[1]Baumgartner, James E., Iterated forcing, Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 159.
[2]Beeson, Michael, The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations, this Journal, vol. 40 (1975), pp. 321346.
[3]Beeson, Michael, Foundations of constructive mathematics, Springer, Berlin, 1985.
[4]Beeson, Michael and Scedrov, Andre, Church's thesis, continuity, and set theory, this Journal, vol. 49 (1984), pp. 630643.
[5]Berger, Josef and Bridges, Douglas, A fan-theoretic equivalent of the antithesis of Specker's Theorem, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae, New Series, vol. 18 (2007), pp. 195202.
[6]Berger, Josef and Bridges, Douglas, The anti-Specker property, a Heine-Borel property, and uniform continuity, Archive for Mathematical Logic, vol. 46 (2008), pp. 583592.
[7]Bishop, Errett, Foundations of constructive mathematics, McGraw-Hill, New York, 1967.
[8]Bishop, Errett and Bridges, Douglas, Constructive analysis, Springer, Berlin, 1985.
[9]Bridges, Douglas, Constructive notions of equicontinuity, Archive for Mathematical Logic, vol. 48 (2009), pp. 437448.
[10]Bridges, Douglas, Inheriting the anti-Specker property, preprint, University of Canterbury, New Zealand, 2009, submitted for publication.
[11]Bridges, Douglas, Uniform equicontinuity and the antithesis of Specker's Theorem, unpublished.
[12]Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vita, Luminita, Strong continuity implies uniformly sequential continuity, Archive for Mathematical Logic, vol. 44 (2005), pp. 887895.
[13]Bridges, Douglas and Richman, Fred, Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987.
[14]Grayson, Robin J., Heyting-valued models for intuitionistic set theory, Applications of sheaves (Fourman, , Mulvey, , and Scott, , editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 402414.
[15]Grayson, Robin J., Heyting-valued semantics, Logic Colloquium'82 (Lolli, , Longo, , and Marcja, , editors), Studies in Logic and the Foundations of Mathematics, vol. 112, North-Holland, Amsterdam, 1984, pp. 181208.
[16]Ishihara, Hajime, Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), pp. 13491354.
[17]Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557565.
[18]Ishihara, Hajime, Sequential continuity in constructive mathematics, Combinatorics, computability, and logic (Calude, , Dinneen, , and Sburlan, , editors), Springer, Berlin, 2001, pp. 512.
[19]Ishihara, Hajime and Schuster, Peter, A continuity principle, a version of Baire's Theorem and a boundedness principle, this Journal, vol. 73 (2008), pp. 13541360.
[20]Ishihara, Hajime and Yoshida, Satoru, A constructive look at the completeness of D(R), this Journal, vol. 67 (2002), pp. 15111519.
[21]Kreisel, Georg, Lacombe, Daniel, and Shoenfield, Joseph, Partial recursive functions and effective operations, Constructivity in mathematics (Heyting, Arend, editor), North-Holland, 1959, pp. 195207.
[22]Lietz, Peter, From constructive mathematics to computable analysis via the realizability interpretation, Ph.D. thesis, Technische Universität Darmstadt, 2004, http://www.mathematik.tu-darmstadt.de/-streicher/THESES/lietz.pdf.gz.
[23]Lubarsky, Robert, Topological forcing semantics with settling, Proceedings of LFCS'09 (Artemov, Sergei N. and Nerode, Anil, editors), Lecture Notes in Computer Science, vol. 5407, Springer, 2009, pp. 309322; also Annals of Pure and Applied Logic, vol. 163 (2012), pp. 820–830.
[24]Lubarsky, Robert, Geometric spaces with no points, Journal of Logic and Analysis, vol. 2 (2010), no. 6, pp. 110.
[25]Lubarsky, Robert and Diener, Hannes, Principles weaker than BD-ℕ, submitted for publication.
[26]Lubarsky, Robert and Rathjen, Michael, On the constructive Dedekind reals, Logic and Analysis, vol. 1 (2008), pp. 131152; also in Proceedings of LFCS'07 (Sergei N. Artemov and Anil Nerode, editors), Lecture Notes in Computer Science, vol. 4514, Springer, 2007, pp. 349–362.
[27]van Oosten, Jap, Extensional realizability, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 317349.
[28]Specker, Ernst, Nicht konstruktiv beweisbare Sätze der Analysis, this Journal, vol. 14 (1949), pp. 145158.
[29]Troelstra, Anne S., A note on non-extensional operations in connection with continuity and recursiveness, Indagationes Mathematicae, vol. 39 (1977), pp. 455462.
[30]Troelstra, Anne S. and van Dalen, Dirk, Constructivism in mathematics, vol. 1, North-Holland, Amsterdam, 1988.
[31]Tseitin, G.S., Algorithmic operators in constructive complete metric spaces, Doklady Akademii Nauk SSSR, vol. 128 (1959), pp. 4952.

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On the failure of BD-ℕ and BD, and an application to the anti-specker property

  • Robert S. Lubarsky (a1)

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