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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

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/0814758371 http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814...

chriswarbo 13y ago
I found it quite approachable and thorough, although it throws the the diagonal lemma out there without any real explanation even though it's crucial to the reasoning that follows it. Also I disagree with the completely unfounded assumptions at the end that the (terribly named) Reals have something to do with reality. The Reals are a Mathematical curiosity at best, but more often than not they complicate the understanding of subjects to which they have no relevance, eg. fractions (especially their decimal notation), calculus, physics, computing (especially floating point) and so on. It's fine to treat infinite constructs like the Reals declaratively, eg. as functions which can be composed, but it's meaningless to reason about doing things to their 'final results', since there are no such things by definition. In more precise terms, it makes sense to reason about the output of co-terminating functions (which loop forever, spitting out, for example, a never-ending sequence of digits) but not diverging functions (which loop forever without ever getting as far as their first digit).
- stiff 13y ago
  The real numbers have no relevance to calculus? Is there a way of doing calculus without real numbers that is simpler? What are you talking about?
hobs 13y ago
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.
- hamiltonkibbe 13y ago
  GEB is awesome. David Foster Wallace's Everything and More: A Compact History of Infinity covers Cantor and is an interesting read about... you guessed it.. Infinity. Re: Turing http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html
- tekromancr 13y ago
  +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.
  - hobs 13y ago
    Yeah it took me about 7 months to finish it, and you are right, every time I just felt beat and my mind was racing, what a great book.
  - laxatives 13y ago
    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 13y ago
      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.
deleted 13y ago
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chriswarbo 13y ago

I found it quite approachable and thorough, although it throws the the diagonal lemma out there without any real explanation even though it's crucial to the reasoning that follows it. Also I disagree with the completely unfounded assumptions at the end that the (terribly named) Reals have something to do with reality. The Reals are a Mathematical curiosity at best, but more often than not they complicate the understanding of subjects to which they have no relevance, eg. fractions (especially their decimal notation), calculus, physics, computing (especially floating point) and so on. It's fine to treat infinite constructs like the Reals declaratively, eg. as functions which can be composed, but it's meaningless to reason about doing things to their 'final results', since there are no such things by definition. In more precise terms, it makes sense to reason about the output of co-terminating functions (which loop forever, spitting out, for example, a never-ending sequence of digits) but not diverging functions (which loop forever without ever getting as far as their first digit).

stiff 13y ago
The real numbers have no relevance to calculus? Is there a way of doing calculus without real numbers that is simpler? What are you talking about?

hobs 13y ago

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.

hamiltonkibbe 13y ago
GEB is awesome. David Foster Wallace's Everything and More: A Compact History of Infinity covers Cantor and is an interesting read about... you guessed it.. Infinity. Re: Turing http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html
tekromancr 13y ago
+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.
- hobs 13y ago
  Yeah it took me about 7 months to finish it, and you are right, every time I just felt beat and my mind was racing, what a great book.
- laxatives 13y ago
  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 13y ago
    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.

deleted 13y ago

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