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Logic Journal of IGPL Advance Access originally published online on February 3, 2009
Logic Journal of IGPL 2009 17(2):173-177; doi:10.1093/jigpal/jzp002
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Indecomposability of negative dense subsets of R in Constructive Reverse Mathematics

Iris Loeb

Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand.
E-mail: I.Loeb{at}math.canterbury.ac.nz

In 1970 Vesley proposed a substitute of Kripke's Scheme. In this paper it is shown that —over Bishop's constructive mathematics— the indecomposability of negative dense subsets of R is equivalent to a weakening of Vesley's proposal. This result supports the idea that full Kripke's Scheme might not be necessary for most of intuitionistic mathematics. At the same time it contributes to the programme of Constructive Reverse Mathematics and gives a new answer to a 1997 question of Van Dalen.

Key Words: Indecomposability • Constructive Reverse Mathematics • Vesley's Substitute • Negative Dense Subsets

Received for publication 17 April 2008.

References

    [1]  Bishop Errett, Douglas Bridges. Constructive Analysis (1985) Springer-Verlag.

    [2]  van Dalen Dirk. How Connected Is the Intuitionistic Continuum? J. Symb. Log (1997) 62(4):1147–1150.[CrossRef]

    [3]  van Dalen Dirk. From Brouwerian Counter Examples to the Creating Subject. Studia Logica (1999) 62(2):305–314.[CrossRef]

    [4]  van Dalen Dirk. Logic and Structure (4th extended, ed.) (2004) Berlin: Springer-Verlag.

    [5]  van Dalen Dirk. How the Mathematical Objects Determine the Mathematical Principles. Journal of Universal Computer Science (2005) 11(12):2132–2141.[Web of Science]

    [6]  Ishihara Hajime. Continuity properties in constructive mathematics. J. Symb. Log. (1992) 57(2):557–565.[CrossRef]

    [7]  Ishihara Hajime. Reverse Mathematics in Bishop's Constructive Mathematics. Philosophia Scientiae, Cahier Spécial 6 (2006) 43–59.

    [8]  Kohlenbach Ulrich. Relative Constructivity. J. Symb. Log (1998) 63(4):1218–1238.[CrossRef]

    [9]  Loeb Iris. Indecompsability of R and R/{0} in Constructive Reverse Mathematics. Logic Journal of IGPL (2008) doi: 10.1093/jigpal/jzn006.

    [10]  Vesley R. E. A Palatable Substitute for Kripke's Schema. In: Intuitionism and Proof Theory: Proceedings of the Summer Conference at Buffalo, N.Y. 1968—Myhill J., Kino A., Vesley R. E., eds. (1970) North-Holland Publishing Company. 197–207.


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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
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Right arrow Articles by Loeb, I.
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What's this?