Completeness for intuitionistic logic. (English) Zbl 0896.03007

McCarty, David

Odifreddi, Piergiorgio (ed.), Kreiseliana: about and around Georg Kreisel. Wellesley, MA: A K Peters. 301-334 (1996).

The author discusses several aspects of intuitionistic completeness, among them the distinction of weak and strong completeness, Gödel’s opinion of intuitionistic incompleteness and topological completeness. He proves some relations between the completeness of Heyting’s predicate logic and Heyting arithmetic on the one hand and Markov’s principle resp. Church’s thesis on the other. The author points out as a central result of his paper “that completeness – even for pure intuitionistic predicate logic – implies Markov’s Principle, operates as a constraint on research into completeness. In the absence of a convincing argument to the conclusion that MP [Markov’s Principle] is intuitionistically incorrect, one is left – as far as mathematics is concerned – with little more than a determination that MP be counted as incorrect and, hence, that full completeness cannot be proved” (p. 331).
For the entire collection see [Zbl 0894.03002].

Reviewer: V.Peckhaus (Erlangen)

MSC:

03A05	Philosophical and critical aspects of logic and foundations
03-03	History of mathematical logic and foundations
03B20	Subsystems of classical logic (including intuitionistic logic)
01A60	History of mathematics in the 20th century
03F55	Intuitionistic mathematics

Keywords:

proof theory; Heyting’s predicate logic; completeness; Markov’s rule; Church’s thesis; intuitionistic incompleteness; lawless sequences; topological models; Heyting arithmetic