4 ms·

What Alan T. did for his PhD

104 points by spottiness 15y ago

frankdilo 15y ago
error establishing db connection
- ionfish 15y ago
  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-independent-of-zf-zfconzf-zfconzfconzfconzf-etc http://mathoverflow.net/questions/67214/pi1-sentence-indepen... [2] http://www.aslonline.org/books-lnl_16.html http://www.aslonline.org/books-lnl_16.html
LukeShu 15y ago
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.
- deleted 15y ago
  [deleted]
- faragon 15y ago
  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.
- davepeck 15y ago
  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-luries-key-insights http://mathoverflow.net/questions/37825/what-are-jacob-lurie...
- gnosis 15y ago
  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
  - Lost_BiomedE 15y ago
    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.
p4bl0 15y ago
Thanks a lot for sharing this link! This kind of things are very interesting to me. To bad for my work...
deleted 15y ago
[deleted]
scythe 15y ago
Related: http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof 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...
radarsat1 15y ago
> 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?
- smanek 15y ago
  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.
  - radarsat1 15y ago
    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?
danbmil99 15y ago
Every time I think I understand how freaking brilliant Alan Turing was, I read shit like this

104 points by spottiness 15y ago

frankdilo 15y ago

error establishing db connection

ionfish 15y ago
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-independent-of-zf-zfconzf-zfconzfconzfconzf-etc http://mathoverflow.net/questions/67214/pi1-sentence-indepen... [2] http://www.aslonline.org/books-lnl_16.html http://www.aslonline.org/books-lnl_16.html

LukeShu 15y ago

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.

deleted 15y ago
[deleted]
faragon 15y ago
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.
davepeck 15y ago
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-luries-key-insights http://mathoverflow.net/questions/37825/what-are-jacob-lurie...
gnosis 15y ago
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
- Lost_BiomedE 15y ago
  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.

p4bl0 15y ago

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

deleted 15y ago

[deleted]

scythe 15y ago

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

radarsat1 15y ago

> 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?

smanek 15y ago
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.
- radarsat1 15y ago
  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?

danbmil99 15y ago

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