Let $\mathcal{N}$ be Peano's 1st order theory of arithmetic and $\mathscr{A}$ it's standard model (which we assume exists). Infer from Gödel's 1st Incompleteness Theorem that there exists a closed well founded formula say $B$ of $\mathcal{N}$ and a model $\mathscr{W}$ of $\mathcal{N}$, such that $B$ is ture in $\mathscr{A}$ and $B$ is false in $\mathscr{W}$. Any help is really needed and app recited. Thanks in advance.

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    $\begingroup$ The theory is usually denoted by $\sf PA$ and the standard model by $\Bbb N$. Also, "wff" means well-formed formula, not well-founded formula. $\endgroup$ – Asaf Karagila Apr 10 '14 at 16:11
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    $\begingroup$ Don't let the words "Peano" and "Gödel" impress you. Being an incomplete theory exactly means to have at least two non elementary equivalent models. $\endgroup$ – Pece Apr 10 '14 at 18:56

Take a canonical Gödel G sentence for PA i.e. a wff that "says" of itself 'I am unprovable in PA'. Ask yourself:

  1. Is G provable from the axioms of PA?
  2. Is G true in every model of the axioms of PA?
  3. Is G true at least in the standard model of PA?
  4. What can you deduce from your last two answers?

Hint: for one of these answers you appeal to Gödel's completeness theorem for first-order logic, and the fact that PA is a first-order theory.

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    $\begingroup$ Picking $G$ is somewhat more specific than you really need. $\endgroup$ – user14972 Apr 10 '14 at 16:39
  • $\begingroup$ Yes, more specific than is needed, but the obvious choice ... $\endgroup$ – Peter Smith Apr 10 '14 at 17:15
  • $\begingroup$ Thanks, Peter This is my reasoning then, would appreciate if you could let me know if it is correct. From the 1st Incompleteness theorem we know we have a sentence $G$ and that $G$ is true, in $\mathbb{N}$ but not provable. So since $G$ is not provable, $PA$ & $\neg G$ is consistent. Therefore by Gödel's completeness theorem, there is a model $\mathscr{W}$ in which $G$ is false. Thanks again really appreciate a great don as yourself answering my question. We used your formal Logic book in my 1st and 2nd years. $\endgroup$ – Jens Apr 10 '14 at 19:14
  • $\begingroup$ Yes indeed. Exactly! (And always good to hear that the Logic book is being used out there!) $\endgroup$ – Peter Smith Apr 10 '14 at 20:32

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