What Alan T. did for his PhD

You know how sometimes on HN there's a conversation that's like<p><pre><code> A&#62; X person is so epic and better than all of us. B&#62; NO, that person is just a normal person like us B&#62; there's nothing special about them. B&#62; We're all capable of being that epic.</code></pre> (a recent one where X="Fabrice Bellard" comes to mind.)<p>The opening paragraph of this does not help person B's argument.

&#62; given any formal system F that we might want to take as a foundation for mathematics (for example, Peano Arithmetic or Zermelo-Fraenkel set theory), Gödel tells us that there are Turing machines that run forever, but that can’t be proved to run forever in F.<p>Wow. I've never understood Godel's theorem before. I've never seen it put that way. Thank you! Is Godel's incompleteness theorem effectively the same thing as the halting problem then? Or rather, a result of it?

Related:<p><a href="http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof" rel="nofollow">http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof</a><p>basically a proof of sorts that Peano arithmetic is consistent based on primitive recursive arithmetic and the "intuitively obvious" idea that there should exist no infinite decreasing sequence of numbers in any set with a minimum element.<p>Also, the consistency of ZFC can be "proven" by proving the existence of a <i>weakly inaccessible cardinal</i>, which is a sort of thing whose existence obviously cannot be proven from within ZFC...

Every time I think I understand how freaking brilliant Alan Turing was, I read shit like this

Thanks a lot for sharing this link! This kind of things are very interesting to me. To bad for my work...

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