Ran Raz: The Surprise Examination Paradox and the Second Incompleteness Theorem

On Wednesday December 5th, there is a talk in the

DIMACS Theory of Computing Seminar which might be
of some interest to logicians:

Place: CORE 431
Time: 11:00 — 12:00

Speaker: Ran Raz (Weizmann Institute & IAS)

Title: The Surprise Examination Paradox and the Second Incompleteness Theorem

Abstract:

Few theorems in the history of mathematics have inspired
mathematicians and philosophers as much as Godel’s first and second
incompleteness theorems. The first incompleteness theorem states that
for any rich enough consistent mathematical theory, there exists a
statement that cannot be proved or disproved within the theory. The
second incompleteness theorem states that for any rich enough
consistent mathematical theory, the consistency of the theory itself
cannot be proved (or disproved) within the theory.
We give a new proof for Godel’s second incompleteness theorem, based
on Kolmogorov complexity and an argument that resembles the surprise
examination paradox.

We then go the other way around and suggest that the second
incompleteness theorem gives a possible resolution of the surprise
examination paradox. Roughly speaking, we argue that the flaw in the
derivation of the paradox is that it contains a hidden assumption that
one can prove the consistency of the mathematical theory in which the
derivation is done; which is impossible by the second incompleteness
theorem.

I will start with a short informal introduction to the known proofs
for Godel’s incompleteness theorems and their relations to the liar
paradox, Kolmogorov complexity and Berry’s paradox.

No knowledge in logic will be assumed.
Joint work with Shira Kritchman

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