How do I best learn Gödel's incompleteness theorem?

Question

I'm a math beginner. I am a self-learner. A while ago, I finished reading “How to Prove It: A Structured Approach, 2nd Edition” by Daniel J. Velleman.

Now, I want to learn Gödel's incompleteness theorem to avoid making the same mistakes made in the past by others.

Where do I find the best learning material for Gödel's incompleteness theorem?

What sort of mistakes do you want to avoid by learning Gödel's incompleteness theorem(s)? — user642796, Oct 07 '15 at 13:34
I'm a programmer. I heard type systems in programming languages are subject to Gödel's incompleteness theorem. I also plan to write programs that deal with sets. Sets may also be subject to Gödel's incompleteness theorem. — , Oct 07 '15 at 13:35
So you actually do not want to know about the incompleteness theorem, but rather how it affects programming? If you want to learn about the theorem, just read about any university logic course book. If you want to learn about the programing part, I am not sure... — Ove Ahlman, Oct 07 '15 at 13:40
This post clearly shows that as long as a formal system can somehow prove the output of any program if it halts, then it must be incomplete. This is simply a matter of string manipulation and not related to sets or types. — user21820, Apr 11 '17 at 05:07
This post gives a more informal proof of a generalized incompleteness theorem based only on knowledge of programming and basic logic. — user21820, Sep 11 '18 at 10:00

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

1

You can see Gödel's Incompleteness Theorems.

Two good books are :

Torkel Franzén, Gödel's theorem : An incomplete guide to its use and abuse (2005)
Peter Smith, An Introduction to Gödel's Theorems (2nd ed - 2013).

And you can see also this post for other references.

edited Apr 13 '17 at 12:21

Community

1

answered Oct 07 '15 at 13:52

Mauro ALLEGRANZA

94,169

score -1 · Answer 2 · answered Oct 07 '15 at 13:42

I learned Gödel's Incompleteness Theorem in high school in the 1980's from two books,

The Lady or the Tiger by Raymond Smullyan http://www.amazon.com/THE-LADY-OR-TIGER/dp/0394514661 and

Gödel, Escher, and Bach by Douglas Hofstadter http://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567

These are entertaining presentations of the theorem in which one doesn't get lost in the details.

If you want to learn more, I would recommend reading Meta-Math by Gregory Chaitin, where he takes things to a higher level. http://arxiv.org/abs/math/0404335

How do I best learn Gödel's incompleteness theorem?

2 Answers2