Is metatheory, providing proof of the incompleteness theorem, consistent? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:04:46Z http://mathoverflow.net/feeds/question/119144 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119144/is-metatheory-providing-proof-of-the-incompleteness-theorem-consistent Is metatheory, providing proof of the incompleteness theorem, consistent? Max 2013-01-17T07:43:42Z 2013-01-17T15:46:13Z <p>Is metatheory, considered as a formal system, that used to prove the First Incompleteness Theorem, consistent?</p> http://mathoverflow.net/questions/119144/is-metatheory-providing-proof-of-the-incompleteness-theorem-consistent/119158#119158 Answer by Steven Landsburg for Is metatheory, providing proof of the incompleteness theorem, consistent? Steven Landsburg 2013-01-17T10:59:08Z 2013-01-17T15:46:13Z <p>Peano Arithmetic suffices to prove Godel's Theorem, and Peano Arithmetic is consistent, so yes.</p> <p>If you don't believe Peano Arithmetic is consistent, then the question becomes: What <em>do</em> you believe is consistent? As you can see in the accepted answer to <a href="http://mathoverflow.net/questions/118183/what-axioms-are-used-to-prove-godels-incompleteness-theorems" rel="nofollow">this question</a>, you can prove Godel's Theorem in <a href="http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic" rel="nofollow">Primitive Recursive Arithmetic</a>, or even a bit less. If you doubt that PRA is consistent, you'll be forced to doubt so much that any skepticism you might have about Godel's Theorem is probably among the least of your worries.</p> http://mathoverflow.net/questions/119144/is-metatheory-providing-proof-of-the-incompleteness-theorem-consistent/119159#119159 Answer by Emil Jeřábek for Is metatheory, providing proof of the incompleteness theorem, consistent? Emil Jeřábek 2013-01-17T11:23:50Z 2013-01-17T11:23:50Z <p>As already pointed out by Steven Landsburg, there are plenty of such theories if you stick to conventional mathematics. If you are some sort of an ultrafinitist, the incompleteness theorem is provable in weak fragments of bounded arithmetic, such as $PV$ or $S^1_2$. These theories are interpretable in Robinson’s arithmetic $Q$, so as long as you accept that $Q$ is consistent, there is such a theory. Do you believe that for every natural numbers $n,m$, the number $\underbrace{2^{2^{\cdot^{\cdot^{\cdot^{2^m}}}}}}_n$ exists? Then $Q$ is consistent.</p>