Reddit mentions of Gödel's Proof

Nagel's book 'Gödel's Proof' is a good, intelligible summary of Gödel. I suggest reading that, even if you suck at math.

• The Elements of Statistical Learning
  
  It's available free online, but I've def got a hard cover copy on my bookshelf. I can't really deal with digital versions of things, I need physical books.
  
  
• If you're looking for something less technical, try The Lady Tasting Tea
  
  
• You haven't mentioned how old your sister is. If she's on the younger side of the spectrum, she might enjoy Flatland.
  
  
• Also, you mention how much your sister loves proofs. Godel's Proof is a really incredible result (sort of brain melting) and the book I linked does a great job of making it accessible. I think I read this book in high school (probably would have understood more if I read it in college, but I got the gist of it).

I really enjoyed Godel's Proof by Nagel + Newman. It's a layman's guide to Godel incompleteness theorem. It avoids some of the more finnicky details, while still giving the overall impression.

https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371/

If you like that, it's edited by Hofstadter, who wrote Godel-Escher-Bach, a famous book about recurrence.

Finally, I would recommend Nonzero: The Logic of Human Destiny by Robert Wright. It's a life-changing book that dives into the relevance of game theory, evolutionary biology and information technology. (Warning that the first 80 pages are very dry.)

https://www.amazon.com/Nonzero-Logic-Destiny-Robert-Wright/dp/0679758941/

Talk about a lack of substance!

A book I read way back when that was excellent was Gödel’s Proof by Nagel and Newman. http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371

A good book on Gödel's proof is Gödel's Proof.

Gödel proved several theorems; I'm guessing you're referring to the incompleteness theorems, which are the most well-known. The key point is that Gödel's incompleteness theorems are precise mathematical statements about certain formal systems — not vague philosophical generalities about the nature of truth or anything like that.

In particular, the content of the first incompleteness theorem is essentially:

>In any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true (in the standard model of arithmetic), but not provable in the theory.

This statement, as with any other statement mathematicians call a "theorem", has been formally proven. Philosophical questions like whether mathematical objects are "real" in whatever sense are irrelevant to the question of whether something is a theorem or not.

By the way, if you want a good introduction to the details of what Gödel's incompleteness theorems say and how they can be proved, I highly recommend Gödel's Proof by Nagel and Newman.

Well I don't know how interested you are in this, but if you want to understand the incompleteness theorem and its implications without learning all of number theory, I ran across this book which provides the history leading up to Gödel, the mathematical context he was working in (e.g. Hilbert's project), and a full explanation of the proof itself in just over 100 pages. I read it in a day, and while I have a background in the area, even if you didn't know anything going into it, you could probably understand the whole thing with two days' careful reading.

I know this is not exactly what you had in mind, but one of the most significant proofs of the 20th century has an entire book written about it:

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

The proof they cover is long and complicated, but the book is nonetheless intended for the educated layperson. It is very, very well written and goes to great lengths to avoid unnecessary mathematical abstraction. Maybe check it out.

If you're looking for a concise introductory level reference, I don't know of any at only the high-school level; additionally most undergrad level textbooks are gonna assume a certain level of sophistication w.r.t. the student.


However, if you are interested, the book "Godel's Proof" by Nagel, offers many accessible insights into the workings of mathemical logic

https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371

>So, my question is- Would you recommend me to skip right into the formal logic parts (and things related, such as computer programs) when reading the book?

I dunno, it depends on what you're trying to get out of the book, I guess. If you just want an exposition of Gödel's incompleteness theorems you can skip to the logic parts, but if that's your goal then there are better books that will get you there faster and more rigorously, like Gödel's Proof by Newman and Nagel, and, incidentally, edited by Hofstadter.

This is the one I read:

http://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371

it's not a bad book but it's got a bad rap

hofstadter writes the foreword to my favorite book on godel's work, this guy

There is a book entitled Godel's proof that was written that was written to explain the ideas of Godel's proof without requiring too much background.

http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?ie=UTF8&qid=1342841590&sr=8-1&keywords=godel%27s+proof

It is hard for me to offer to much advice beyond that because I am in a different field of mathematics (number theory).

For something more rigorous than "Godel, Escher, Bach" try "Godel's Proof" by Nagel and Newman.

>Can they be shown to be consistent?

Not by the metamathematics itself, no. It's a result from Gödel's Incompleteness Theorems that no consistent mathematical system that can be mapped into arithmetics can demonstrate it's own consistency.

This book does a good job of explaining Gödel's work, you should consider reading it.

Highly recommend Godel's Proof for anybody looking to jump into the question of how well founded modern mathematics is.

Agree. I picked up on that from the intro to GEB, stopped reading GEB, and decided to get a better understanding of Gödel's proof by reading the book Hofstadter says introduced him to Gödel - Gödel's Proof, by Ernest Nagel and James R. Newman. I recommend it as a very approachable introduction to Gödel's incompleteness theorems. Even now I can recall moments reading that little book where I'd get a big smile on my face as the force of his argument and conclusion would bear down on me. What Gödel did is nothing short of mind blowing.

After that, if you want more, then go to Gödel's Incompleteness Theorems by Raymond M. Smullyan (You'll want to buy this one used). This one is a much more technical, though still approachable if you're prepared at an undergrad level, guide through to Gödel's conclusions. You should go into it with an undergrad level of fluency in propositional and predicate logic.

You can read GEB without all that, certainly without the second book, but I've found it a better experience having more familiarity with Gödel as I work through it.

Book recommendation for an intro to Godel's Theorem: Gödel's Proof - Ernest Nagel and James Newman. Well written, concise and requires no prior mathematical knowledge.

Edit: Never mind. misread "I do have an introductory understanding..." as "I don't have an introductory understanding...". Still a good recommendation for anyone else who is interested!

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

160 pages:

https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?ie=UTF8&qid=1486067268&sr=8-1&keywords=godels+proof

Gödel's Proof is a good starting point for the incompleteness theorem. Covers the basics of the theorem and its impacts. Unless you are prepping for coursework in logic than this book likely has the right amount of depth for you.

I don't have a recommendation for Tarski. Hopefully someone else has something for you.

I was totally enthralled with the philosophy of Mathematics when I was in college. One of the books I found interesting -- before I had progressed in mathematical logic -- was this one on Godel's Proof.

this book is quite short but perfect for an aspiring mathematician that is going to start hearing about Gödel's proof in casual conversation. This provides a concise easy treatment of it's importance and how the proof works. Also, see it's reviews on goodreads

There's a great introduction to Gödel's Incompleteness Theorems, it's called and Gödel's Proof by Nagel & Newman. Hofstadter has wrote it's foreword. It's a very short book, 160 pages in total.

Amazon Link!

BTW, if you want a relatively easy description of Godel's work, this book may be useful.

I am a Strange Loop is about the theorem

Another book I recommend is David Foster-Wallace's Everything and More. It's a creative book all about infinity, which is a very important philosophical concept and relates to mind and machines, and even God. Infinity exists within all integers and within all points in space. Another thing the human mind can't empirically experience but yet bears axiomatic, essential reality. How does the big bang give rise to such ordered structure? Is math invented or discovered? Well, if math doesn't change across time and culture, then it has essential existence in reality itself, and thus is discovered, and is not a construct of the human mind. Again, how does logic come out of the big bang? How does such order and beauty emerge in a system of pure flux and chaos? In my view, logic itself presupposes the existence of God. A metaphysical analysis of reality seems to require that base reality is mind, and our ability to perceive and understand the world requires that base reality be the omniscient, omnipresent mind of God.

Anyway these books are both accessible. Maybe at some point you'd want to dive into Godel himself. It's best to listen to talks or read books about deep philosophical concepts first. Jay Dyer does a great job on that

https://www.youtube.com/watch?v=c-L9EOTsb1c&t=11s

Those are/were my interests and I enjoyed [Logical Dilemmas] (http://www.amazon.com/Logical-Dilemmas-Life-Work-Godel/dp/1568812566/ref=sr_1_1?ie=UTF8&qid=1324312359&sr=8-1), a thorough biography of Kurt Godel. [Godel's Proof] (http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?s=books&ie=UTF8&qid=1324312476&sr=1-1) might be too basic but is a good read.

Godel's Proof is the original inspiration for Hofstadter. I find it a shorter but no less interesting read.

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

When I first went through it, I found it very verbose and too abstract for me. I was clearly not prepared for it.

Then I happened to read Gödel's proof, by Nagel and Newman, with an updated commentary by Hofstader. What a terrific book! Having gone through it, I began enjoying GEB.

There's tremendous depth in both books, and I look forward to iterating through these two alternately and getting more and more insights.