Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T12:28:42.344Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  12 March 2014

Robert S. Lubarsky
Robert S. Lubarsky*
Affiliation:
Dept. of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Anti-SpeckerBDBD-ℕrealizabilitytopological models
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 39 - 56
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baumgartner, James E., Iterated forcing, Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 1–59.Google Scholar
[2]Beeson, Michael, The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations, this Journal, vol. 40 (1975), pp. 321–346.Google Scholar
[3]Beeson, Michael, Foundations of constructive mathematics, Springer, Berlin, 1985.CrossRefGoogle Scholar
[4]Beeson, Michael and Scedrov, Andre, Church's thesis, continuity, and set theory, this Journal, vol. 49 (1984), pp. 630–643.Google Scholar
[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. 195–202.Google Scholar
[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. 583–592.CrossRefGoogle Scholar
[7]Bishop, Errett, Foundations of constructive mathematics, McGraw-Hill, New York, 1967.Google Scholar
[8]Bishop, Errett and Bridges, Douglas, Constructive analysis, Springer, Berlin, 1985.CrossRefGoogle Scholar
[9]Bridges, Douglas, Constructive notions of equicontinuity, Archive for Mathematical Logic, vol. 48 (2009), pp. 437–448.CrossRefGoogle Scholar
[10]Bridges, Douglas, Inheriting the anti-Specker property, preprint, University of Canterbury, New Zealand, 2009, submitted for publication.Google Scholar
[11]Bridges, Douglas, Uniform equicontinuity and the antithesis of Specker's Theorem, unpublished.Google Scholar
[12]Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vita, Luminita, Strong continuity implies uniformly sequential continuity, Archive for Mathematical Logic, vol. 44 (2005), pp. 887–895.CrossRefGoogle Scholar
[13]Bridges, Douglas and Richman, Fred, Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[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. 402–414.CrossRefGoogle Scholar
[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. 181–208.CrossRefGoogle Scholar
[16]Ishihara, Hajime, Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), pp. 1349–1354.Google Scholar
[17]Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557–565.Google Scholar
[18]Ishihara, Hajime, Sequential continuity in constructive mathematics, Combinatorics, computability, and logic (Calude, , Dinneen, , and Sburlan, , editors), Springer, Berlin, 2001, pp. 5–12.CrossRefGoogle Scholar
[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. 1354–1360.Google Scholar
[20]Ishihara, Hajime and Yoshida, Satoru, A constructive look at the completeness of D(R), this Journal, vol. 67 (2002), pp. 1511–1519.Google Scholar
[21]Kreisel, Georg, Lacombe, Daniel, and Shoenfield, Joseph, Partial recursive functions and effective operations, Constructivity in mathematics (Heyting, Arend, editor), North-Holland, 1959, pp. 195–207.Google Scholar
[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.Google Scholar
[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. 309–322; also Annals of Pure and Applied Logic, vol. 163 (2012), pp. 820–830.Google Scholar
[24]Lubarsky, Robert, Geometric spaces with no points, Journal of Logic and Analysis, vol. 2 (2010), no. 6, pp. 1–10.CrossRefGoogle Scholar
[25]Lubarsky, Robert and Diener, Hannes, Principles weaker than BD-ℕ, submitted for publication.Google Scholar
[26]Lubarsky, Robert and Rathjen, Michael, On the constructive Dedekind reals, Logic and Analysis, vol. 1 (2008), pp. 131–152; 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.CrossRefGoogle Scholar
[27]van Oosten, Jap, Extensional realizability, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 317–349.CrossRefGoogle Scholar
[28]Specker, Ernst, Nicht konstruktiv beweisbare Sätze der Analysis, this Journal, vol. 14 (1949), pp. 145–158.Google Scholar
[29]Troelstra, Anne S., A note on non-extensional operations in connection with continuity and recursiveness, Indagationes Mathematicae, vol. 39 (1977), pp. 455–462.CrossRefGoogle Scholar
[30]Troelstra, Anne S. and van Dalen, Dirk, Constructivism in mathematics, vol. 1, North-Holland, Amsterdam, 1988.Google Scholar
[31]Tseitin, G.S., Algorithmic operators in constructive complete metric spaces, Doklady Akademii Nauk SSSR, vol. 128 (1959), pp. 49–52.Google Scholar