Godel's Theorem
by Torkel Franzen · first published 2005
6mentions
4recommended by
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2010–2022first – last mentioned
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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.
on “Set Theory and Foundations of Mathematics” ↗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.
on “The Danger When You Don't Know What You Don't Know” ↗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
on “A Python implementation of formal systems from the book “Gödel, Escher, Bach”” ↗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…
on “Memories of Kurt Gödel” ↗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).
on “Gödel's Incompleteness Theorem and Its Implications for Artificial Intelligence” ↗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.
on “Incompleteness ex Machina – proving Gödel's theorems in terms of algorithms [pdf]” ↗