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Summary: Understanding Constructive Semantics
(Spinoza Lecture)
Sergei N. Artemov
Cornell and Moscow University
http://www.cs.cornell.edu/home/artemov
August 17, 1999
Abstract
Is there an alternative mathematics? In particular, does intuitionism yield an essen-
tially new approach that cannot be speci ed within classical mathematics? The intended
informal meaning of intuitionistic logic Int was given in the 1930s by the Brouwer-Heyting-
Kolmogorov semantics which understands intuitionistic truth as provability. Moreover,
Kolmogorov (and later Godel) suggested interpreting Int via classical provability and
thus providing a meaningful semantics for Int independent of intuitionistic assumptions.
Natural attempts to formalize this semantics met serious diĘculties related to Godel's
incompleteness phenomenon. In this lecture we will talk about recent advances in this
area that have bridged the incompleteness gap and provided an adequate formalization of
the propositional Brouwer-Heyting-Kolmogorov semantics based on classical provability.
Plan
1. Brouwer - Heyting - Kolmogorov provability semantics for intuitionistic logic
2. De ning intuitionistic logic in classical provability logic
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