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Suppose I am working in a standard formal theory such as ZFC or NBG.

Consider this statement:

"For all well-formed formulas s, ((there exists a proof p s.t. p is a valid proof of s) --> s)"

I'm fairly familiar with some of the basic facts of logic, e.g., Tarski's undefinability theorem, Godel's theorems, and the difference between "consistency" and "soundness."

Nonetheless, I have two questions:

1) Is this a theorem of ZFC/NBG? (What would a proof look like?)

2) Is there a term for this property of formal theories? E.g., "soundness," "consistency," and "completeness" are terms for other different properties.

  • I'm pretty sure this cannot even be stated in the language of ZFC or NBG - you cannot express truth of arbitrary sentence in the language. – Wojowu Apr 14 '17 at 19:53
  • I think you are right. Technically, I would have to say, "For all integers n in N, ... --> n is the Godel number of a true sentence." Thanks. –  Apr 14 '17 at 19:59
  • That still won't work - one can't express "n is the Godel number of a true sentence". – Wojowu Apr 14 '17 at 20:00
  • I understand...I was just explaining why it would not work. –  Apr 14 '17 at 20:02
  • Ah, I see. Yes, that's correct. However, you can still ask about systems which prove, for every sentence, "s is provable -> s". – Wojowu Apr 14 '17 at 20:03
  • Actually, on second thought, I think your initial comment was mistaken...see the answer below. –  Apr 14 '17 at 20:16
  • The answer below (and Lob's theorem itself) doesn't deal with actual truth of statements, but with provability. To rephrase it in your terms, Lob's theorem gives ""For all well-formed formulas s, (Provable(there exists a proof p s.t. p is a valid proof of s) --> Provable(s))". – Wojowu Apr 14 '17 at 20:19
  • @Wojowu: Löb's theorem does not say that. Read my answer and the link therein. For a sound system, like PA, Löb's theorem implies that a system like ZFC cannot prove $\mathsf{Provable}(\phi) \to \phi$ for any $\phi$ that is not true. – Rob Arthan Apr 14 '17 at 21:46
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    @RobArthan Ah, I see my mistake. I will have to give this thing a better look later. Thanks for clarifying. – Wojowu Apr 14 '17 at 21:48
  • @Wojowu: For much more details, see this post. – user21820 Apr 15 '17 at 06:21

1 Answers1

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Löb's theorem says that (in any sufficiently strong system) for any sentence $\phi$, if $\mathsf{Provable}(\phi) \to \phi\;$ is provable then $\phi$ is provable (so that the hypothesis in $\mathsf{Provable}(\phi) \to \phi\;$ was vacuous). A sound system that proves $\mathsf{Provable}(\phi) \to \phi\;$ for any $\phi$ that is not true is inconsistent.

Rob Arthan
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