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text: '[summary: In theories extending [-minimal_arithmetic] there exists a $\\exists_1$-formula $\\square_T$(x) which defines the existence of a proof in an [ axiomatizable theory] $T$. \n\nThe formula satisfies [5j7 some nice propereties], but fails to capture some intuitions about provability due to non-standard weirdness.]\n\nWe know that every theory as [expressiveness_mathematics expressive] as [-minimal_arithmetic] (i.e., [Peano_Arithmetic $PA$]) is capable of talking about [statement_mathematics statements] about itself via [encoding encodings] of [sentence_mathematics sentences] of the language of [-arithmetic] into the [45h natural numbers].\n\nOf particular interest is figuring out whether it is possible to write a formula $Prv(x)$ which is true [-46m] there exists a proof of $x$ from the axioms and rules of inference of our theory.\n\nIf we reflect on what a proof is, we will come to the conclusion that a proof is a sequence of symbols which follows certain rules. Concretely, it is a sequence of strings such that either:\n\n1. The string is an [-axiom] of the system or...\n2. The string is a theorem that can be deduced from previous strings of the system by applying a [ rule of inference].\n\nAnd the last string in the sequence corresponds to the theorem we want to prove.\n\nIf the axioms are a [-computable_set], and the rules of inference are also effectively computable, then checking whether a certain string is a proof of a given theorem is also computable.\n\nSince every computable set can be defined by a [ $\\Delta_1$-formula] in arithmetic %%note:[ Proof]%% then we can define the $\\Delta_1$-formula $Prv(x,y)$ such that $PA\\vdash Prv(n,m)$ iff $m$ encodes a proof of the sentence of arithmetic encoded by $n$.\n\nThen a sentence $S$ is a theorem of $PA$ if $PA\\vdash \\exists y Prv(\\ulcorner S \\urcorner, y)$.\n\nThis formula is the standard provability predicate, which we will abbreviate as $\\square_{PA}(x)$, and has some nice properties which correspond to what one would expect of a [-5j7].\n\nHowever, due to [non_standard_model non-standard models], there are some intuitions about provability that the standard provability predicate fails to capture. For example, it turns out that $\\square_{PA}A\\rightarrow A$ [55w is not always a theorem of $PA$].\n\nThere are non-standard models of $PA$ which contain numbers other than $0,1,2,..$ (called [non_standard_models_of_arithmetic non-standard models of arithmetic]). In those models, the standard predicate $\\square_{PA}x$ can be true even if for no natural number $n$ it is the case that $Prv(x,n)$. %%note:This condition is called [ $\\omega$-inconsistency]%% This means that there is a non-standard number which satisfies the formula, but nonstandard numbers do not encode standard proofs!',
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