Godel's Theorem

What Hacker News says

1. wrpSep 2022
   An informative overview which is not referenced in the article is Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. (https://www.goodreads.com/book/show/12830.G_del_s_Theorem).
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2. jimhefferonAug 2017
   Lots of reading are available to technical people who are willing. People elsewhere in this thread have mentioned Nagel and Newman, and also Smullyan. Here are two more:

   1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy.

   2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf is enlightening in a different way.

   I'll also mention a favorite author around here http://www.scottaaronson.com/blog/?p=710 .

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3. jimhefferonMar 2017
   I'm not sure why you have it in quotes. But FWIW, most of my questions were cleared up by these two wonderful books: Peter Smith's An Introduction to Gödel's Theorems and Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse
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4. pjungwirJul 2015
   I felt similar amazement at Gödel's proof, and even suspect that Penrose must be on to something when he says the mind must therefore not be computable. . . . But the book Gödel's Theorem: An Incomplete Guide to its Use and Abuse by Törkel Franzén is very good at explaining that the conclusions we should draw are fairly limited, e.g. that the proof doesn't apply to anything other than arithmetic. I'm not sure I'm totally convinced, but it's a great perspective to dispel some of the more mystical interpretations.
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5. qwzybugMay 2010
   Gödel's theorem in particular lends itself to this sort of thing. For more mathematical schadenfreude, I like Franzen's "An Incomplete Guide to its Use and Abuse" http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/...

   Hackers will be interested especially in the rather weak attempts to use Gödel to make all sorts of silly inferences about artificial intelligence and the nature of mind.

   Probably no scientific result of the 20th centry, except maybe the theory of relativity, has been used by so many to prove so much while understanding so little.

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6. soberhoffMar 2019
   I don't agree with that definition of "effectively axiomatized". Granted, that's what Wikipedia says here: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... But the reference given doesn't check out. Franzén's 2005 book Gödel's Theorem doesn't say anything about effective axiomatization on page 112. He does however give the following definition in his lecture notes at: https://books.google.de/books?id=Wr51DwAAQBAJ&pg=PA138#v=one...

   "If T is effectively axiomatized, in the sense that there is an algorithm for deciding whether or not a given sentence is an axiom of T,..."

   In Computability: Turing, Gödel, Church and Beyond Martin Davis even makes the definition "T is axiomatizable if it has an axiom set that is computable" on page 42.

   And that's also the definition Shoenfield gives on page 125 of Mathematical Logic.

   Besides, "effective" has always been a synonym for "computable", "recursive", or "decidable" whenever I've encountered it in this context.

   Also, I must insist that "it makes no difference whether we require the axioms to be recursively enumerable, recursive, or primitive recursive." is not what Craig's Theorem says. This is dangerously close to claiming that recursively enumerable sets are recursive which is plainly untrue. If recursively enumerable axioms (not just recursively enumerable theorems) are indeed interchangeable with recursive axioms, then I'd like to hear more about that.

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