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Gödel's arithmetization of logic is one hell of a beautiful mathematical idea, up there with linear algebra or probability theory. One of its big selling points
by cousin_it 10y ago
Gödel's arithmetization of logic is one hell of a beautiful mathematical idea, up there with linear algebra or probability theory. One of its big selling points for me was how easily it solves the unexpected hanging paradox. (The simplest arithmetization of the judge's self-referential statement is a statement about numbers that can be shown to be self-contradictory.) It's also cool how it turns Russell's paradox into Gödel's theorem, Curry's paradox into Löb's theorem, etc. The connections to algorithms and computability theory are also neat (Gödel's idea of "effectively axiomatized system" is any computer program that can print sentences, which has just the right amount of generality and connects to the halting problem in the obvious way). To me arithmetization is simply the right approach to logic, which easily subsumes everything that makes sense and rejects everything that doesn't.
- westoncb 10y agoDo you have a recommendation for a book that covers these topics that's technical but not necessarily a text book?
- empath75 10y agoThe classic popular treatment is: Godel Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter
- westoncb 10y agoI've read about a third of it. It's too long and too focused on other topics for me personally. I also read the book 'Gödel's Proof' a long time ago, before I had much familiarity with pure math, and remember it describing a very interesting sort of 'architecture' that Gödel had put together—but I left still wondering about any applications other than producing the incompleteness theorems. Maybe I'd get more out of it now though...
- alasdair_ 10y agoGEB is well worth a second attempt. It's one of the very few books that completely changed my worldview.
- westoncb 10y agoI lost interest when certain other ideas made me think his basic project of accounting for consciousness mathematically was deeply flawed (and it wasn't Penrose's argument about formal systems!). I did find it entertaining though, so maybe I'll revisit before too long. I'm curious what shift it caused for you, if you don't mind saying.
- empath75 10y agoAt least for me, it wasn't so much about computers or computer science, as it was the first time I really deeply considered what consciousness is. It's actually the book, along with The Selfish Gene that made me dismiss the necessity of any kind of 'soul', fresh out of Catholic School.
- westoncb 10y agoThe more time I've spent reading about and considering the nature of subjectivity, the more I believe arguments from evolution or computer simulation are irrelevant. They're talking about something different.
- fenomas 10y agoI read GEB a few years ago, and it struck me as a book full of ideas that would have transformed me had I been encountering them for the first time. I think the book may have been so influential that its big ideas had largely filtered out into places where I'd encountered them. If I'd read it at 19, say, I think it would have rocked my world. Note that I'm mostly talking about the book's first half (about incompleteness and canons and whatnot). The second half (ants, etc) struck me as much more dated.
- aromatt 10y agoThere's a chapter about it in _I Am a Strange Loop_ (Douglas Hofstadter) which I found very enjoyable and pretty enlightening.
- GregBuchholz 10y agoI liked "Godels Theorem Simplified". Although it looks like it is now a lot more expensive than it used to be. Maybe get it through and interlibrary loan. https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensler/dp/081913869X https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensl...
- cousin_it 10y agoI really like Peter Smith's "Gödel Without Tears" lecture notes: http://www.logicmatters.net/resources/pdfs/gwt/GWT2f.pdf http://www.logicmatters.net/resources/pdfs/gwt/GWT2f.pdf
- njohnson41 10y agoI personally liked "Godel's Proof" by Ernest Nagel. It's pretty old, but it's to-the-point and inexpensive.