The double negation of the intermediate value theorem |
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Authors: | Mohammad Ardeshir Rasoul Ramezanian |
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Institution: | Department of Mathematics, Sharif University of Technology, P. O. Box 11365-9415, Tehran, Iran |
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Abstract: | In the context of intuitionistic analysis, we consider the set F consisting of all continuous functions ? from 0,1] to R such that ?(0)=0 and ?(1)=1, and the set I0 consisting of ?’s in F where there exists x∈0,1] such that . It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer’s continuity principle we have I0≠F. However, there exists no satisfying answer to . We try to answer to this question by reducing it to a schema (which we call ) about intuitionistic decidability that asserts “there exists an intuitionistically enumerable set that is not intuitionistically decidable”. We also introduce the notion of strong Specker double sequence, and prove that the existence of such a double sequence is equivalent to the existence of a function ?∈Fmon where . |
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Keywords: | 03F55 03F60 |
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