{"product":{"product_id":"0814758371","title":"Gödel's Proof","price":"13.39","image_url":"https://m.media-amazon.com/images/I/41tf1Cd6IkL._SL500_.jpg","url":"https://www.amazon.com/dp/0814758371"},"comments":[{"body":"There is an excellent brief introduction to the topic called *Gödel's Proof* by Nagel and Newman. It's only a bit over 150 pages (including a forward and appendix) and, as such, does not go into exacting detail, but it is sufficient to give you a working understanding of why the proof is important, how it works, where the pitfalls arise, etc. It does assume a knowledge of basic mathematical logic (a first undergraduate course in logic would be more than sufficient preparation), but that's about it. It's a pretty easy read written in a conversational style.\r\n\r\n\r\nIt's currently about $10 USD on Amazon: https://www.amazon.com/dp/0814758371/\r\n\r\nThere's also a pdf floating around, if you are so inclined.","subreddit_name":"math","author":"smuttyinkspot"},{"body":"Godel is not a threat because his proof demonstrates that Russell's project in Principia Mathematica is not feasible. When set theory arrived in mathematics the quest was to build a rigorous system based off a finite number of axioms. Godel proved this is impossible. It is a statement about set theory and the limits in constructing axiomatic systems to include all mathematical truths, namely that such a venture is impossible. Check out Nagel's book [Godel's Proof](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?crid=PDZ45YTDNYUX\u0026amp;dchild=1\u0026amp;keywords=godel%27s+proof+by+ernest+nagel+and+james+r.+newman\u0026amp;qid=1609809886\u0026amp;sprefix=nagel+godel%2Caps%2C179\u0026amp;sr=8-1). It makes a rather difficult work understandable. Godel's actual paper is very hard to follow.","subreddit_name":"CatholicPhilosophy","author":"pinkfluffychipmunk"},{"body":"The book \"[Gödel's Proof](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/)\" does a good job explaining it without getting *too* much in the technicals.  If you *really* wanna understand it, I'd recommend it.  You could probably get through that book in a day if you really tried, and you'd definitely walk away understanding how it works (at least at a high-level).","subreddit_name":"mathmemes","author":"Nater5000"},{"body":"There's a great introduction to Gödel's Incompleteness Theorems, it's called and Gödel's Proof by Nagel \u0026amp; Newman. Hofstadter has wrote it's foreword. It's a very short book, 160 pages in total.\r\n\r\n[Amazon Link!](http://www.amazon.com/dp/0814758371)","subreddit_name":"zen","author":"GodVonGodel"},{"body":"You deleted another comment, I think, but I wrote up another reply, and I thought it might help.\r\n\r\nYou can do multiplication by adding a specific number of times in Presburger arithmetic, but you can't do multiplication of some variable by another. I.e., the following are allowed:\r\n\r\n1. a + b\r\n2. a + 2 * b\r\n3. 8353 * a + 12853 * b\r\n\r\nBut 'a * b' is not allowed. The other multiplications you can unroll into repeated additions, but multiplication by a variable here requires an axiom (or several) from Peano arithmetic to give the 'rules' for what multiplication in general means: https://en.wikipedia.org/wiki/Peano_axioms#Multiplication\r\n\r\nWhy does this solve the issue? Well, Gödel's proof relies on using the properties of Peano numbers to encode a way of talking about statements and their proofs using the numbers themselves. And in this way, he comes up with a statement that basically means 'there is no proof of this statement', which leads to the consequences of his theorem. But one of these properties that he relies on is (iirc) a way of constructing unique representations of statements and proofs using powers of prime numbers. If you want to read more about this, I recommend [Gödel's proof](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371).\r\n\r\nSo you can see that if multiplication is dropped, Gödel's proof is no longer possible. However, this on its own wouldn't necessarily mean it had to be consistent and complete, but it was proved to be so by Presburger (although I am not familiar with the proof of it).","subreddit_name":"math","author":"turingchurch"},{"body":"Hofstadter was cribbing (as he acknowledged later, I believe in *I am an Infinite Loop*) from [this book by Nagel](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371).","subreddit_name":"slatestarcodex","author":"SandyPylos"},{"body":"I think this is one of the better books that goes in-depth about Gödel's proofs with pretty much no prerequisites.\r\n\r\n[https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371)","subreddit_name":"math","author":"Richard_Fey"},{"body":"Godel's incompleteness theorem is without a doubt the most beautiful theorem out there imo. Nagel and Newman's [Godel's Proof](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371) was a fantastic read","subreddit_name":"wallstreetbetsHUZZAH","author":"Deaths_Intern"},{"body":"Excellent article! \r\n\r\nSmall correction: it's [Nagel and Newman](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371) and not \"Nadel and Newman\" as stated at the bottom.","subreddit_name":"Clojure","author":"orbiscerbus"}]}