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SEPARATING THE FAN THEOREM AND ITS WEAKENINGS

Published online by Cambridge University Press:  18 August 2014

ROBERT S. LUBARSKY  and
HANNES DIENER
ROBERT S. LUBARSKY
Affiliation:
DEPT. OF MATHEMATICAL SCIENCES FLORIDA ATLANTIC UNIVERSITY BOCA RATON, FL 33431, USAEmail: [email protected]
HANNES DIENER
Affiliation:
DEPARTMENT MATHEMATIK, FAK. IV EMMY-NOETHER-CAMPUS, WALTER-FLEX-STR. 3 UNIVERSITY OF SIEGEN 57068 SIEGEN, GERMANYEmail: [email protected]
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Abstract

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

Keywords

fan theoremsKripke modelsforcingnon-standard modelsHeyting-valued models03C9003F5003F6003H05
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 3 , September 2014 , pp. 792 - 813
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Beeson, Michael, Foundations of Constructive Mathematics,Springer-Verlag / Berlin, Germany, 1985.CrossRefGoogle Scholar
Berger, Josef, The logical strength of the uniform continuity theorem, Logical Approaches to Computational Barriers, Proceedings of CiE 2006, LNCS 3988, 2006, pp. 35–39.CrossRefGoogle Scholar
Berger, Josef, A separation result for varieties of Brouwer’s Fan Theorem, Proceedings of the 10th Asian Logic Conference (ALC 10), Kobe University in Kobe, Hyogo, Japan, September 1–6, 2008, 2010, pp. 85–92.Google Scholar
Bishop, Errett and Bridges, Douglas, Constructive Analysis, Springer-Verlag / Berlin, Germany, 1985.Google Scholar
Diener, Hannes, Compactness under constructive scrutiny, Ph.D. thesis, 2008.Google Scholar
Diener, Hannes and Loeb, Iris, Sequences of real functions on [0, 1] in constructive reverse mathematics. Annals of Pure and Applied Logic, vol. 157 (2009), no. 1, pp. 50–61.Google Scholar
Fourman, M. and Hyland, J., Sheaf models for analysis, Applications of Sheaves (Fourman, Michael, Mulvey, Christopher, and Scott, Dana, editors.) Lecture Notes in Mathematics, vol. 753, Springer Berlin / Heidelberg, 1979, pp. 280–301,Google Scholar
Julian, William and Richman, Fred, A uniformly continuous function on [0,1] that is everywhere different from its infimum. Pacific Journal of Mathematics, vol. 111 (1984), no. 2, pp. 333–340,Google Scholar
Lubarsky, Robert S., Independence Results around Constructive ZF, Annals of Pure and Applied Logic, vol. 132 (2005), no. 2–3, pp. 209–225,Google Scholar
Simpson, Stephen, Subsystems of Second Order Arithmetic, ASL/Cambridge University Press / Cambridge, UK, 2009.Google Scholar