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text: '[summary: If $PA\\vdash Prv_{PA}(A)\\implies A$ then $PA\\vdash A$]\n\n\n\nWe trust Peano Arithmetic to correctly capture certain features of the [ standard model of arithmetic]. Furthermore, we know that Peano Arithmetic is expressive enough to [31z talk about itself] in meaningful ways. So it would certainly be great if Peano Arithmetic asserted what now is an intuition: that everything it proves is certainly true.\n\nIn formal notation, let $Prv$ stand for the [-5gt] of $PA$. Then, $Prv(T)$ is true if and only if there is a proof from the axioms and rules of inference of $PA$ of $T$. Then what we would like $PA$ to say is that $Prv(S)\\implies S$ for every sentence $S$.\n\nBut alas, $PA$ suffers from a problem of self-trust.\n\nLöb's theorem states that if $PA\\vdash Prv(S)\\implies S$ then $PA\\vdash S$. This immediately implies that if $PA$ is consistent, the sentences $PA\\vdash Prv(S)\\implies S$ are not provable when $S$ is false, even though according to our intuitive understanding of the standard model every sentence of this form must be true.\n\nThus, $PA$ is incomplete, and fails to prove a particular set of sentences that would increase massively our confidence in it.\n\nNotice that [godels_second_incompleteness_theorem Gödel's second incompleteness theorem] follows immediately from Löb's theorem, as if $PA$ is consistent, then by Löb's $PA\\nvdash Prv(0= 1)\\implies 0= 1$, which by the propositional calculus implies $PA\\nvdash \\neg Prv(0= 1)$.\n\nIt is worth remarking that Löb's theorem does not only apply to the standard provability predicate, but to every predicate satisfying the [ Hilbert-Bernais derivability conditions].',
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