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Tait’s conservative extension theorem revisited. (English) Zbl 1190.03015

In “The completeness of Heyting first-order logic” [J. Symb. Log. 68, No. 3, 751–763 (2003; Zbl 1055.03036)], W. W. Tait claims the conservativeness of CH over HL. Here, CH is the Tait version of Curry-Howard construction theory and HL is Heyting (i.e. intuitionistic) logic. In this paper, the author points out that Tait’s proof of the main theorem [Theorem 5.1] is defective, and hence the conservativeness above, which is its corollary, is not yet proved. Even worse, by providing a counter-example, he shows that Theorem 5.1 is just plain wrong. However, the author amends Tait’s proof, shows the CH/HL conservativeness, and presents a modified version of Theorem 5.1. He exhibits a few examples which are totally lacking in Tait’s article, but repeats definitions and explains again mechanisms and meanings of various notions.
In item [1] of the references, p. 167, “pp. 221–224” should be “pp. 127–172”.

MSC:

03B20 Subsystems of classical logic (including intuitionistic logic)
03F25 Relative consistency and interpretations
03F65 Other constructive mathematics

Citations:

Zbl 1055.03036
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References:

[1] The provenance of pure reason: Essays in the philosophy of mathematics and its history (2005) · Zbl 1111.00006
[2] The completeness of Hey ting first-order logic 68 pp 751– (2003)
[3] Twenty-five years of constructive type theory pp 221– (1998)
[4] DOI: 10.1007/BF00413978
[5] Journal of Philosophical Logic 12 pp 173– (1983)
[6] Mathematics and mind pp 45– (1994) · Zbl 0834.00012
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