What Alan T. did for his PhD | Hacker News Reader

What Alan T. did for his PhD

scottaaronson.com

104 pointsspottiness13 comments

You know how sometimes on HN there's a conversation that's like

  A> X person is so epic and better than all of us.
  B> NO, that person is just a normal person like us
  B> there's nothing special about them.
  B> We're all capable of being that epic.

(a recent one where X="Fabrice Bellard" comes to mind.)

The opening paragraph of this does not help person B's argument.

Fabrice Bellard is clearly far above the average geek. Probably he is not even aware of it. Other examples that in my opinion can be identified clearly and distinctly (just naming few): Bertrand Russell, Alan Turing, Henri Poincaré.

We talk of 10x, but the truth is we are rarely confronted with it in our daily lives. Maybe 1.5x, but not 10x. When we do meet 10x, the order of magnitude is so staggering that we attempt to explain it away.

Here's another 10x, who happened to go to my high school: http://mathoverflow.net/questions/37825/what-are-jacob-lurie...

  If I have seen further it is only by standing on the shoulders of giants.

            -- Isaac Newton

  The great appear great in our eyes
  Only because we are kneeling.
  Let us rise!

            --Elisée Loustalot

The Isaac Newton quote is pretty funny. Robert Hooke claimed that Newton's work copied his own. Newton wrote the quote as an insult when replying to Hooke, who was very short.

The original variation, not Newton, did have the expected meaning, though.

> 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.

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?

The halting problem is a specific instance of the type of problems predicted by the incompleteness theorem (IT).

The first IT says there within any system of logic that's powerful enough to express arithmetic (and consistent), there are always statements that are true that can't be proved true. A specific program, P, that doesn't halt, but can't be proved not to halt, is an example of this. (Or, more precisely, the statement 'The program P halts' is the example.)

The second IT says you can't prove the consistency of a system from within that system itself, but that's another story.

Okay, so then you can't say from some point of view that IT is a _result_ of the halting problem, right? It's only the other way around?

http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof

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.

Also, the consistency of ZFC can be "proven" by proving the existence of a weakly inaccessible cardinal, 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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All the details are on MathOverflow [1], and the recommendation of Franzén's Inexhaustibility [2] is spot on.

[1] http://mathoverflow.net/questions/67214/pi1-sentence-indepen...

[2] http://www.aslonline.org/books-lnl_16.html

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