2022-11-30: Fredrik Nordvall Forsberg (MSP)

Sat, 21 Oct 2023 21:24:08 +0100

Fredrik Nordvall Forsberg (MSP)

Brouwerian counterexamples: being a know-it-all 101

Abstract

Some statements are not provable, in the sense that they imply falsity (for example: "every inhabited subset of real numbers has a least upper bound"). Other statements are not provable constructively, even though they are true when using classical logic (for example: "every bounded inhabited subset of real numbers has a least upper bound"). How can we convince ourselves of this fact, so that we do not waste time and effort looking for a constructive proof? We can construct a so-called Brouwerian counterexample by showing that the claimed statement implies a known non-constructive principle. Brouwer called such principles omniscience principles, because only an all-knowing being can hope to for example decide if P or not P holds for every proposition P. I will give an introduction to some of the more common omniscience principles such as LPO, LLPO and Markov's Principle, and try to give some intuition for when one should expect a statement to not have a constructive proof; an important skill in order to avoid heartbreak when working in a constructive setting such as type theory or the internal logic of a category.

2022-11-30 11:00:00 UTC
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