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# What is Mathematics: Gödel's Theorem and Around

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Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. CONTENTS. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem. [[[[[[[]]]]]]] Russian version available: https://www.researchgate.net/publication/306112090_Around_Godel%27s_Theorem_2nd_edition_in_Russian [[[[[[[]]]]]]] For Part 1 see https://www.researchgate.net/publication/349104699_Introduction_to_Mathematical_Logic_Edition_2021
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... This allows us to represent the proposition " T is consistent " by some arithmetical formula Con(T ). If Con(T ) is built in the most direct way, then one can prove in T itself that, indeed, Con(T ) expresses the consistency of T (for technical details, if needed, see K. Podnieks (2015)). It appears, that the formula Con(T ) → G T can then be formally proved in T. Thus, if one assumes that T is consistent, i.e., if one assumes Con(T ), then proving the formula G T does not require specific " human powers " ; one can use the axioms of T instead! ...
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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.
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This is OLD Edition 2017. NEW Edition 2021 is available at https://www.researchgate.net/publication/349104699_Introduction_to_Mathematical_Logic_Edition_2021.
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