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What is Mathematics: Gödel's Theorem and Around
Abstract
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
... 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! ...
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.
This chapter formulates seven long-standing problems related to consciousness using introspection and a certain amount of experimental evidence provided by biomedical sciences. The physical boundary problem is to find a rule that sets the boundary between our own conscious mind and the rest of the physical world. The binding problem is to explain what binds our conscious experiences into a single whole. The causal potency problem is to explain how our mind could act upon the physical world. The free will problem is to explain how it is possible for us to make genuine choices between two or more alternative future courses of action. The inner privacy problem is to explain why we have a privileged access to our unobservable conscious minds whose phenomenal content is incommunicable to others. The mind-brain relationship problem is to explain whether the mind and the brain differ, and if they do, how they interact with each other. The hard problem of consciousness is to explain how our brain generates consciousness and why we have any conscious experiences at all.
This book addresses the fascinating cross-disciplinary field of quantum information theory applied to the study of brain function. It offers a self-study guide to probe the problems of consciousness, including a concise but rigorous introduction to classical and quantum information theory, theoretical neuroscience, and philosophy of the mind. It aims to address long-standing problems related to consciousness within the framework of modern theoretical physics in a comprehensible manner that elucidates the nature of the mind-body relationship. The reader also gains an overview of methods for constructing and testing quantum informational theories of consciousness.
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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