G\"{o}del's incompleteness theorem and the Anti-Mechanist Argument: revisited

This is a paper for a special issue of the journal “Studia Semiotyczne” devoted to Stanislaw Krajewski’s paper [30]. This paper gives some supplementary notes to Krajewski’s [30] on the Anti-Mechanist Arguments based on G"{o}del’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4-6 in Krajewski’s [30] and classify some misunderstandings of G"{o}del’s incompleteness theorem related to Anti-Mechanist Arguments. In Section 4 and 5, we give a more detailed discussion of G"{o}del’s Disjunctive Thesis, G"{o}del’s Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7-8 in Krajewski’s [30], describing how recent advances bear on these issues.