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text: '[summary: Every [ consistent] and [ axiomatizable] extension of [ minimal arithmetic] is incomplete]\n\nThe **first incompleteness theorem** is a result about the existence of sentences of arithmetic that cannot be proved or disproved, no matter what axioms we take as true.\n\nWhat Gödel originally proved was that every [ $\\omega$-consistent] and [ axiomatizable] extension of Peano Arithmetic is incomplete, but the result was later refined to weaken the requirement of $\\omega$-consistency to simple [-5km] and the set of theorems that the extension had to prove to that of [ minimal arithmetic].\n\nThe heart of both proofs is the [-59c], which allows us express self-referential sentences in the language of arithmetic.\n\nThis put an end to the dream of building a [-complete] [-5hh] that axiomatized all of mathematics, since as soon as one was expressive enough to talk about arithmetic, incompleteness would kick in.\n\n##Interpretation from model theory\nThe first incompleteness theorem highlights the impossibility of defining the [45h natural numbers] with the usual operations of [ addition] and [ multiplication] in [ first order logic].\n\nWe already knew from Lowenhëim-Skolem theorem that there would be models of $PA$ which are not isomorphic to the usual arithmetic, but the first incompleteness theorem implies that some of those models disagree in the truth value of some theorems of this language (those are the undecidable sentences).',
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