

Gödel, Turing and Cantor: The Math - jamesmiller5
http://skibinsky.com/godel-turing-and-cantor-the-math/

======
wicknicks
Oh wow! This is a great write up on Godel's work. Anybody who even vaguely
cares about fundamentals of computer science should definitely give it a read,
and if possible a thorough read.

Slightly related: Although a more technical/deeper discussion, but the book
"Godel's Proof"[1] by Nagel and Newman is a very approachable text in this
domain, and explains many aspects of the incompleteness theorems.

[1] [http://www.amazon.com/G%C3%B6dels-Proof-Ernest-
Nagel/dp/0814...](http://www.amazon.com/G%C3%B6dels-Proof-Ernest-
Nagel/dp/0814758371)

~~~
hobs
Honorable mention for
[http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach](http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach)
which has been mentioned many times on HN. Has many different ways of
explaining Godel's genius. Though to be fair you could just read this article
and be good.

~~~
tekromancr
+1 for GEB. I have been reading it on and off for about 2 years. It's a fun
book, but every time I finish a reading session, I am exhausted. I am about
25% through.

~~~
laxatives
IMO just finish chapter 14 and you've basically finished the Godel portion of
the book. The rest felt more like Hofstadter's random musings on AI, DNA and
some other topics (some of which felt outdated). There's some crazy anecdotes
of Ramanujin that I had never heard elsewhere though.

~~~
pachydermic
Well the structure of the book is supposed to be like a JS Bach song - so in
the end, what you get is a combining of all the familiar themes and sometimes
it feels a bit repetitive and boring. But if you've read that far, it's worth
reading the rest. It's much easier to get through than when you're seeing
stuff for the first time and really trying to wrap your head around it.

------
6ren
There's a translation of Gödel's paper linked via his wikipedia entries
[https://docs.google.com/viewer?url=http://www.research.ibm.c...](https://docs.google.com/viewer?url=http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf)
[google docs viewer]

curious facts: he died of starvation; he was a theist
[http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#Later_years_and...](http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#Later_years_and_death)

~~~
kybernetikos
I found your semicolon quite confusing there - it seemed to suggest that there
was some relationship between being a theist and dying of starvation.

~~~
Radim
On the contrary.

I imagine 6ren went to extra lengths to use a semicolon instead of a simple
comma, as semicolons are used to emphasize independent clauses.

~~~
kybernetikos
Independent clauses can stand by themselves _grammatically_ , that doesn't
mean that their meanings are independent.

e.g.

> I hit him as hard as I could; he laughed at me.

Those are 'independent clauses', but the semicolon emphasizes the connection
between them.

~~~
6ren
Radim is right. I actually tried a couple of other forms ("and" and ","), but
they seemed worse. Challenge: can you demonstrate a better way?

I think the problem is not syntax but semantics: two unusual facts beg
connection.

BTW: getting back on topic, his starvation seems due to a lack of faith... if
some connection is expected, perhaps the best solution is to supply one,
perhaps the poignant _He believed in God, yet died from lack of faith._

------
mvp
This article took me to another from the same author

'[http://skibinsky.com/godel-incompleteness-for-
startups/#foot...](http://skibinsky.com/godel-incompleteness-for-
startups/#footnote-4')

and I found the below statement and laughed out aloud. Although in the
context, this statement makes sense, in general, I trust books that don't
belong to human-related matters (technical etc), because in human-related
matters it is mostly one's opinion against others.

"As soon as these popular books leave the domain of human-related matters you
are totally on your own."

------
Cacti
Chaitin is cool, too.

~~~
andyjohnson0
Definitely. I'd recommend his book _Meta Math: The Quest for Omega_ [1]. He
posted the full text of the book on arXiv [2].

[1] [https://www.amazon.co.uk/Meta-Maths-Gregory-J-
Chaitin/dp/184...](https://www.amazon.co.uk/Meta-Maths-Gregory-J-
Chaitin/dp/184354525X) [2]
[http://arxiv.org/abs/math/0404335](http://arxiv.org/abs/math/0404335)

------
bitdiddle
What I find interesting are the claims Penrose[1] and others make about human
versus machine intelligence based on Godel's results.

[1]
[https://en.wikipedia.org/wiki/Philosophy_of_artificial_intel...](https://en.wikipedia.org/wiki/Philosophy_of_artificial_intelligence#Lucas.2C_Penrose_and_G.C3.B6del)

~~~
chriswarbo
I'm sceptical of Penrose's arguments. We don't see human brains as symbol-
manipulating formal systems, but that doesn't mean that aren't (or, more
correctly, that doesn't mean that there is no symbol-manipulating formal
system which is isomorphic to a particular brain, or brains in general).

Brains aren't implementing any known algorithm, but that doesn't mean they
aren't implementing any unknown algorithm.

It brings to mind Minksky's advice to Sussman
[http://en.wikipedia.org/wiki/Hacker_koan#Uncarved_block](http://en.wikipedia.org/wiki/Hacker_koan#Uncarved_block)

------
stanonyime
It would be interesting to speculate the consequence of Godel's Incompleteness
deductions on the quest for Grand Unified Field Theorems in theoretical
physics

------
brisance
Does this mean that the antivirus industry is nothing but a scam?

~~~
lordmax
it means antivirus industry will never be "complete"

~~~
chriswarbo
Indeed, many incompleteness results (the most famous being the Halting
Problem) are known as "infinite employment theorems". For example, we can
always make a compiler perform more optimisations as follows:

1) Every diverging (non-terminating, non-co-terminating) program can be
optimised to the following:

10 GOTO 10

2) The Halting Problem tells us that no compiler can spot every diverging
program.

3) Hence there is room for more optimisation, by spotting more diverging
programs.

4) Hence there is always more work to do for compiler developers.

