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  text: 'A consistent [-theory] is one in which there are well-formed statements that you cannot prove from its axioms; or equivalently, that there is no $X$ such that $T\\vdash X$ and $T\\vdash \\neg X$.\n\nFrom the point of view of [-model_theory], a consistent theory is one whose axioms are [-satisfiable]. Thus, to prove that a set of axioms is consistent you can resort to constructing a model using a formal system whose consistency you trust (normally using [set_theory]) in which all the axioms come true.\n\nArithmetic is [ expressive enough] to talk about consistency within itself. If $\\square_{PA}$ represents the [-5gt] in [3ft] then a sentence of the form $\\neg\\square_{PA}(\\ulcorner 0=1\\urcorner)$ represents the consistency of $PA$, since it comes to say that there exists a disprovable sentence for which there is no proof. [ Gödel's second incompleteness theorem] comes to say that such a sentence is not provable from the axioms of $PA$.',
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