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Principles Weaker than BD-N

  • Robert S. Lubarsky (a1) and Hannes Diener (a2)


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.



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[1] Beeson, Michael, The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations, this Journal, vol. 40 (1975), pp. 321346.
[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. 195202.
[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. 583592.
[4] Berger, Josef and Bridges, Douglas, Rearranging series constructively, Journal of Universal Computer Science, vol. 15 (2009), pp. 31603168.
[5] Berger, Josef, Bridges, Douglas, Diener, Hannes, and Schwichtenberg, Helmut, Constructive aspects of Riemann's permutation theorem for series, submitted for publication.
[6] Berger, Josef, Bridges, Douglas, and Palmgren, Erik, Double sequences, almost Cauchyness, and BD-N, Logic Journal of the IGPL, vol. 20 (2012), pp. 349354.
[7] Bridges, Douglas, Constructive notions of equicontinuity, Archive for Mathematical Logic, vol. 48 (2009), pp. 437448.
[8] Bridges, Douglas, Inheriting the anti-Specker property, Documenta Mathematica, vol. 15 (2010), pp. 9073–980.
[9] Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vîţǎ, Luminita, Strong continuity implies uniformly sequential continuity, Archive for Mathematical Logic, vol. 44 (2005), pp. 887895.
[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. 402414.
[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. 181208.
[12] Ishihara, Hajime, Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), pp. 13491354.
[13] Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557565.
[14] Ishihara, Hajime, Sequential continuity in constructive mathematics, Combinatorics, Computability, and Logic (Calude, , Dinneen, , and Sburlan, , editors), Springer, 2001, pp. 512.
[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. 13541360.
[16] Ishihara, Hajime and Yoshida, Satoru, A constructive look at the completeness of d(r), this Journal, vol. 67 (2002), pp. 15111519.
[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.
[18] Lubarsky, Robert, Geometric spaces with no points, Journal of Logic and Analysis, vol. 2 (2010), no. 6, pp. 110.
[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. 3956.



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