The cause of axiomatic development was taken up in earnest by the Bourbaki group of mathematicians. Taken to its extreme this attitude demanded developing mathematics in its greatest generality, starting from the most general axioms and then specializing (e.g. introducing vector spaces over arbitrary fields and limiting to the real numbers only when absolutely necessary) even when the specializations were the theorems of primary interest. In particular, this perspective placed little value on fields of mathematics (such as combinatori ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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Gödel's theorems are theorems in first-order logic, and must ultimately be understood in that context. In formal logic, both mathematical statements and proofs are written in a symbolic language, one where we can mechanically check the validity of proofs so that there can be no doubt that a theorem follows from our starting list of axioms. In theory, such a proof can be checked by a computer, and in fact there are computer programs that will check the validity of proofs. (Automatic proof verification is closely related to automated theo ...

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Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes

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Gödel's theorems are theorems in first-order logic, and must ultimately be understood in that context. In formal logic, both mathematical statements and proofs are written in a symbolic language, one where we can mechanically check the validity of proofs so that there can be no doubt that a theorem follows from our starting list of axioms. In theory, such a proof can be checked by a computer, and in fact there are computer programs that will check the validity of proofs. (Automatic proof verification is closely related to automated theo ...

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Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes

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In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Axiom - Logical axioms. These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all ...

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Axiom, Axiom - Etymology, Axiom - Mathematics, Axiom - Logical axioms, Axiom - Non-logical axioms, Axiom - Role in mathematical logic, Axiom - Further discussion

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List of mathematics lists - Basic mathematics. These lists cover topics typically taught in secondary education or in the first year of university. List of basic discrete mathematics topics List of calculus topics List of geometry topics List of topics in logic List of trigonometry topics < ...

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List of mathematics lists, List of mathematics lists - Fields of mathematics, List of mathematics lists - Basic mathematics, List of mathematics lists - Advanced mathematics, List of mathematics lists - Applied mathematics, List of mathematics lists - Methodology, List of mathematics lists - Mathematical statements, List of mathematics lists - General concepts, List of mathematics lists - Mathematical objects, List of mathematics lists - About mathematics, List of mathematics lists - Mathematicians, List of mathematics lists - Trivia, List of mathematics lists - Reference tables, List of mathematics lists - Integrals, List of mathematics lists - Meta-lists, List of mathematics lists - Notes

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List of mathematics lists - Integrals. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. The following pages list the integrals of many different functions. List of integrals List of integrals of arc functions List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals ...

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List of mathematics lists, List of mathematics lists - Fields of mathematics, List of mathematics lists - Basic mathematics, List of mathematics lists - Advanced mathematics, List of mathematics lists - Applied mathematics, List of mathematics lists - Methodology, List of mathematics lists - Mathematical statements, List of mathematics lists - General concepts, List of mathematics lists - Mathematical objects, List of mathematics lists - About mathematics, List of mathematics lists - Mathematicians, List of mathematics lists - Trivia, List of mathematics lists - Reference tables, List of mathematics lists - Integrals, List of mathematics lists - Meta-lists, List of mathematics lists - Notes

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List of mathematics lists - Mathematicians. Mathematicians study and research in all the different areas of mathematics. The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics). List of geometers List of logicians List of mathematicians List of probabilist ...

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List of mathematics lists, List of mathematics lists - Fields of mathematics, List of mathematics lists - Basic mathematics, List of mathematics lists - Advanced mathematics, List of mathematics lists - Applied mathematics, List of mathematics lists - Methodology, List of mathematics lists - Mathematical statements, List of mathematics lists - General concepts, List of mathematics lists - Mathematical objects, List of mathematics lists - About mathematics, List of mathematics lists - Mathematicians, List of mathematics lists - Trivia, List of mathematics lists - Reference tables, List of mathematics lists - Integrals, List of mathematics lists - Meta-lists, List of mathematics lists - Notes

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