
What Alan T. did for his PhD - spottiness
http://www.scottaaronson.com/blog/?p=697
======
LukeShu
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.

~~~
gnosis

      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
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.

------
radarsat1
> 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
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
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?

------
scythe
Related:

<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...

------
danbmil99
Every time I think I understand how freaking brilliant Alan Turing was, I read
shit like this

------
p4bl0
Thanks a lot for sharing this link! This kind of things are very interesting
to me. To bad for my work...

------
frankdilo
error establishing db connection

~~~
ionfish
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...](http://mathoverflow.net/questions/67214/pi1-sentence-independent-
of-zf-zfconzf-zfconzfconzfconzf-etc)

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

