Michał Tomasz Godziszewski
(University of Warsaw)
Computational properties of undecidable sentences and concrete model theory
March 27th at 17:30, in F1.15 ILLC seminar room
Emergence and rapid development of recursion theory and computer science enable us to rigorously address the question of characterising the class of mathematical concepts that are cognitively accessible to computational devices such as human minds. The answer to this question would give reasons for which some concepts are epistemically easy (e.g. provable within first-order theories or possessing certain combinatorial properties) and the others are cognitively hard for the human mind. We justify a methodological stance that mathematical knowability may be identified with algorithmic learnability. We present the framework of experimental logics that is equivalent to the notion of learnability. Then we answer the question of computational reasons for epistemic hardness of certain philosophically interesting mathematical concepts. Specifically, we give an example - by adjoining the minimal possible set of undecidable sentences to recursive axiomatization of arithmetics and closing it under logical consequence, we obtain a non-learnable theory.