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Principles Weaker than BD-N

Published online by Cambridge University Press:  12 March 2014

Robert S. Lubarsky  and
Hannes Diener
Robert S. Lubarsky
Affiliation:
Dept. of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA, E-mail: Robert.Lubarsky@alum.mit.edu
Hannes Diener
Affiliation:
Department Mathematik, Fak. IV, Emmy-Noether-Campus, Walter-Flex-Str. 3, University of Siegen, 57068 Siegen, Germany, E-mail: diener@math.uni-siegen.de
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Abstract

BD-N is a weak principle of constructive analysis. Several interesting principles implied by BD-N have already been identified, namely the closure of the anti-Specker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to be strictly weaker than BD-N, yet not provable in set theory alone under constructive logic.

Keywords

03F6003F5003C9026E40Anti-Specker spacesBD-NCauchy sequencespartially CauchyRiemann Permutation Theoremtopological models
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 873 - 885
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] 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
[2] 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
[3] 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.Google Scholar
[4] Berger, Josef and Bridges, Douglas, Rearranging series constructively, Journal of Universal Computer Science, vol. 15 (2009), pp. 3160–3168.Google Scholar
[5] Berger, Josef, Bridges, Douglas, Diener, Hannes, and Schwichtenberg, Helmut, Constructive aspects of Riemann's permutation theorem for series, submitted for publication.Google Scholar
[6] Berger, Josef, Bridges, Douglas, and Palmgren, Erik, Double sequences, almost Cauchyness, and BD-N, Logic Journal of the IGPL, vol. 20 (2012), pp. 349–354.Google Scholar
[7] Bridges, Douglas, Constructive notions of equicontinuity, Archive for Mathematical Logic, vol. 48 (2009), pp. 437–448.Google Scholar
[8] Bridges, Douglas, Inheriting the anti-Specker property, Documenta Mathematica, vol. 15 (2010), pp. 9073–980.Google Scholar
[9] Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vîţǎ, Luminita, Strong continuity implies uniformly sequential continuity, Archive for Mathematical Logic, vol. 44 (2005), pp. 887–895.CrossRefGoogle Scholar
[10] Grayson, Robin J., Heyting-valued models for intuitionistic set theory, Applications of sheaves (Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 402–414.Google Scholar
[11] Grayson, Robin J., Heyting-valued semantics, Logic Colloquium '82 (Lolli, G., Longo, G., and Marcja, A., editors), Studies in Logic and the Foundations of Mathematics, vol. 112, North-Holland, Amsterdam, 1984, pp. 181–208.Google Scholar
[12] Ishihara, Hajime, Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), pp. 1349–1354.Google Scholar
[13] Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557–565.Google Scholar
[14] Ishihara, Hajime, Sequential continuity in constructive mathematics, Combinatorics, Computability, and Logic (Calude, , Dinneen, , and Sburlan, , editors), Springer, 2001, pp. 5–12.Google Scholar
[15] 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
[16] Ishihara, Hajime and Yoshida, Satoru, A constructive look at the completeness of d(r), this Journal, vol. 67 (2002), pp. 1511–1519.Google Scholar
[17] Lietz, Peter, From Constructive Mathematics to Computable Analysis via the Realizability Interpretation, Ph.D. thesis, Technische Universität Darmstadt, 2004, see also Realizability models refuting Ishihara's boundedness principle, joint with Thomas Streicher, submitted for publication.Google Scholar
[18] Lubarsky, Robert, Geometric spaces with no points, Journal of Logic and Analysis, vol. 2 (2010), no. 6, pp. 1–10.Google Scholar
[19] Lubarsky, Robert, On the failure of BD-ℕ and BD, and an application to the anti-Specker property, this Journal, vol. 78 (2013), no. 1, pp. 39–56.Google Scholar