Decidability and Specker sequences in intuitionistic mathematics

Mathematical Logic Quarterly 55 (6):637-648 (2009)

Abstract

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema about intuitionistic decidability that asserts “there exists an intuitionistic enumerable set that is not intuitionistic decidable” and show that the existence of a Specker sequence is equivalent to ED. We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them

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References found in this work

Nicht Konstruktiv Beweisbare Sätze der Analysis.Ernst Specker - 1949 - Journal of Symbolic Logic 14 (3):145-158.
An Interpretation of Intuitionistic Analysis.D. van Dalen - 1978 - Annals of Mathematical Logic 13 (1):1.
Specker Sequences Revisited.Jakob G. Simonsen - 2005 - Mathematical Logic Quarterly 51 (5):532-540.

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Citations of this work

The Double Negation of the Intermediate Value Theorem.Mohammad Ardeshir & Rasoul Ramezanian - 2010 - Annals of Pure and Applied Logic 161 (6):737-744.
Compactness, Colocatedness, Measurability and ED.Mohammad Ardeshir & Zahra Ghafouri - 2018 - Logic Journal of the IGPL 26 (2):244-254.
On the Constructive Notion of Closure Maps.Mohammad Ardeshir & Rasoul Ramezanian - 2012 - Mathematical Logic Quarterly 58 (4-5):348-355.

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