# What technical books should I read to understand clearly Gödel's theorems and their implications for math?

What technical books should I read to understand clearly Gödel's theorems and their implications for math?

I had a course in university which covered set theory, relations, functions, cardinality (with also Cantor's diagonal argument), boolean algebra viewed as a distributive lattice and Stone's theorem (without the proof), sentential calculus and first order logic (defined with syntax and semantic), tableaux method for both, Hilbert system and deduction theorem (and so, the sequent calculus). Also, without proving it, the professor said that there is a demonstration of soundness and completeness of the Hilbert system (all tautologies and only them can be derived).

After the course I read "Gödel's proof" by Nagel and Newmann and "The universal computer" by Martin Davis. Now I want to understand more. I want to understand how logic can embrace every mathematical reasoning, the difference between an axiomatic theory and a formal system and how a formal system can be built from an axiomatic theory (if it makes sense... maybe I only have confused ideas). And of course i would like to understand the real Gödel's incompleteness theorem proof without someone to simplify it for me (like in Nagel and Newmann's book).

I have to agree with the wise words of Mauro Allegranza :-)

If you want quite a bit of detail in a reasonably accessible form -- and free! -- then there's my notes Godel Without (Too Many Tears).

If you want the same basic approach developed more discursively in book form, then there is my Introduction to Gödel's Theorems. (The 2nd edition is much better than the 1st in the early chapters which relate to your worries, so do look out for that.)

But different people of course like different presentations, so I'd also warmly recommend the books by Epstein and Carnielli, and by Boolos, Burgess and Jeffrey, that you can find details about in §4.2 of this Guide to logic books.

This is a really fun area of logic where you get to Big And Important Results surprisingly quickly and easily. So enjoy!

• Hi Peter - I find your comments throughout MSE helpful and appreciate their succinct nature. You probably have noticed that I have asked you several times for you own opinion regarding the relative math/logic contents of your IGT and WTs versions. Of course, I have looked at both and have no doubt they merit the high regard many commenters throughout MSE have expressed. In your remarks above, you do characterize IGT as more discursive. I did look up 'discursive.' Ironically it has two definitions: 1) tending to digress from the main point, and 2) using reason and argument rather than (cont.) – user12802 Jun 19 '18 at 14:18
• intuition. So perhaps you would give me some resolve. I definitely want to read one of the presentations. If I have conviction that IGT has substantially more math/logic content (for the material where they overlap), I will gladly pursue that with focus. Thanks, – user12802 Jun 19 '18 at 14:19
• IGT indeed has substantially more math/logic content -- it is discursive in the sense of having more chat about the mathematical results too. – Peter Smith Jun 19 '18 at 14:47