A note on applicability of the incompleteness theorem to human mind

AbstractWe shall present some relations between consistency and reflection principles which explain why is Gödel's incompleteness theorem wrongly used to argue that thinking machines are impossible

On the depth of G\"{o}del's incompleteness theorem

In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of G\"{o}del's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with arXiv:2009.0488

Gödel\u27s Incompleteness Theorems

The Incompleteness Theorems of Kurt Godel are very famous both within and outside of mathematics. They focus on independence and consistency within mathematics and hence a more thorough understanding of these is beneficial to their study. The proofs of the theorems involve many ideas which may be unfamiliar to many, including those of formal systems, Godel numbering, and recursive functions and relations. The arguments themselves mirror the Liar’s Paradox in that Godel constructs a statement asserting its own unprovability and then shows that such a statement and its negation must both be independent of the system, otherwise the system is inconsistent. We then proceed to survey various interpretations of the Incompleteness Theorems, focusing on potential misapplications of the theorems

Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics

The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or were neglected in past discussions

On the philosophical relevance of Gödel's incompleteness theorems

A survey of more philosophical applications of Gödel's incompleteness results