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  text: '##Weak form\nThe weak Gödel's first incompleteness theorem states that every [ $\\omega$-consistent] [ axiomatizable] extension of minimal arithmetic is incomplete.\n\nLet $T$ extend [-minimal_arithmetic], and let $Prv_{T}$ be the [5gt standard provability predicate] of $T$. \n\nThen we apply the [59c diagonal lemma] to get $G$ such that $T\\vdash G\\iff \\neg Prv_{T}(G)$.\n\nWe assert that the sentence $G$ is undecidable in $T$. We prove it by contradiction:\n\nSuppose that $T\\vdash G$. Then $Prv_ {T}(G)$ is correct, and as it is a $\\exists$-rudimentary sentence then it is [every_true_e_rudimentary_sentence_is_provable_in_minimal_arithmetic provable in minimal arithmetic], and thus in $T$. So we have that $T\\vdash Prv_ {T}(G)$ and also by the construction of $G$ that $T\\vdash \\neg Prv_{T}(G)$, contradicting that $T$ is consistent.\n\nNow, suppose that $T\\vdash \\neg G$. Then $T\\vdash  Prv_{T}(G)$. But then as $T$ is consistent there cannot be a standard proof of $G$, so if $Prv_{T}(x)$ is of the form $\\exists y Proof_{T}(x,y)$ then for no natural number $n$ it can be that $T\\vdash Proof_ {T}(\\ulcorner G\\urcorner,n)$, so $T$ is $\\omega$-inconsistent, in contradiction with the hypothesis.\n\n##Strong form\n\n> Every [5km consistent] and [-axiomatizable] extension of [-minimal_arithmetic] is [complete incomplete].\n\nThis theorem follows as a consequence of the [ undecidability of arithmetic] combined with the lemma stating that [ any complete axiomatizable theory is undecidable]\n',
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