Gödel's Incompleteness Theorem states that in any non-contradictory non-trivial axiomatic system there are certain statements or theorems whom cannot be proved in that system,

For example in ZFC, assuming it is non-contradictory, there is the famed Continuum Hypothesis and Whitehead's problem. My question is, is there any such system (but different from ZFC) in which Gödel's Incompleteness Theorem itself is an unprovable statement?

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    $\begingroup$ Are you sure that the Riemann Hypothesis is unprovable in ZFC (if ZFC is consistent)? You should write that up for publication -- you'll be famous! $\endgroup$ – hmakholm left over Monica Aug 19 '12 at 14:51
  • $\begingroup$ This is not really about set theory as much as it is about logic and incompleteness. I retagged it appropriately. (And it's not very soft either) $\endgroup$ – Asaf Karagila Aug 19 '12 at 14:52
  • $\begingroup$ @HenningMakholm Ah yes you're right. Then it means that I probably read about another famous theorem that was proved unprovable within ZFC. And Asaf, yes, you're right, thanks, and I originally tagged it as soft as I did not expect a proof or anything very concrete, given the question asked. $\endgroup$ – andreas.vitikan Aug 19 '12 at 14:59
  • $\begingroup$ @andreas.vitikan: The most famous statements that are proved independent of ZFC are the Continuum Hypothesis and the Generalized Continuum Hypothesis. Since those have "hypothesis" in their names, they might be what you're thinking about. $\endgroup$ – hmakholm left over Monica Aug 19 '12 at 15:01
  • $\begingroup$ @Henning Ah yes, that was it, thanks for spotting it! $\endgroup$ – andreas.vitikan Aug 19 '12 at 15:03

Robinson arithmetic is non-trivial enough that the incompleteness theorem applies to it, but as far as I know not strong enough to prove the incompleteness theorem itself.


Just a footnote to Henning Makholm's answer. You indeed need to look at very weak systems for examples of the phenomenon you want. For a start, as soon as you add a little bit of induction to Robinson Arithmetic (officially, induction for $\Sigma_1$ wffs) you get a theory that, for any recursively axiomatized $T$ which includes a bit of arithmetic and any sensible system of Gödel-coding, can show that $Con_T \to \neg Prov_T(\overline{\ulcorner G_T\urcorner)}$, i.e. it can prove the formalized version of (half of) Gödel's incompleteness theorem for $T$. (It was rather important to Gödel that the incompleteness theorem is elementary in this sort of way.)


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