2

I want to fully understand Godel's incompleteness theorem.

my background knowledge are these: analysis1, linear algebra1,2 , abstract algebra1, topology1

and I studied logic by myself while reading this book 'introduction to logic, Irving Copi'

I also read the book 'Gödel's Proof Book by Ernest Nagel and James R. Newman'

but I think this book is not good enough and it has some errors

I wish I could read the original Godel's paper but I think there are some prerequisites which I should learn.

so what steps do I have to take to fully understand godel's theorem?

I think it would be good to recommend books to read in order.

Mauro ALLEGRANZA
  • 94,169
ju so
  • 287
  • I have also read Nagel Newman's book and found it exceptional. What are your concerns? – Hanno Aug 07 '21 at 05:19
  • 2
    My recommendation on this topic is: "An introduction to Gödel's theorems" by Peter Smith. – Léreau Aug 07 '21 at 08:01
  • 1
    Also SEP's entry is quite detailed. – Mauro ALLEGRANZA Aug 07 '21 at 09:17
  • I suggest do an exercise (or study until you can) prove the uniqueness of prime factorization in robinson arithmetic+induction. – DanielV Aug 08 '21 at 09:36
  • 3
    Considering your mathematical background you could go straight to Raymond Smullyan's book 'Gödel's Incompleteness Theorems'. If you feel you need some recursion theory or basic logic beforehand consult Neil Cutland's 'Computability' and Herbert Enderton's 'A Mathematical Introduction to Logic'. Both of these texts cover the first incompleteness theorem as well. A more advanced monograph on the incompleteness theorems is Torkel Franzen's 'Inexhaustibility: A Non-Exhaustive Treatment – sequitur Aug 08 '21 at 23:16
  • You could consider reading A Friendly Introduction to Mathematical Logic by Leary. If you already know you want to go more deeply into logic, then you could read the book by Cori and Lascar instead, or even the book by Hinman. – Anonymous Aug 23 '21 at 16:32
  • 1
    Although I first learnt the incompleteness theorems from Peter Smith's book, in my personal opinion it is not as good as the computability-based proof. That linked thread not only is quite self-contained but also proves a full generalization of the theorem, and best of all fits in a single SE post. With your background, you can fully understand the theorem (first half of the post) within a day, and if you need clarification on the extras (second half) feel free to ask me! – user21820 Nov 27 '22 at 14:16

1 Answers1

3

I'd second @Lereau's recommendation of An introduction to Gödel's theorems by Peter Smith. What a surprise!

You can download a PDF from https://www.logicmatters.net/igt (though the pbk is very cheap, if like many people you prefer to work from a physical book).

But you can also find other recommendations -- with descriptions of how they approach the topic -- in Chapter 8 of the Beginning Mathematical Logic Study Guide which you can download from https://www.logicmatters.net/tyl.

Peter Smith
  • 54,743
  • thanks for your comment! I am considering both your book and Smullyan's book. – ju so Aug 09 '21 at 10:23
  • so I asked my school library to buy both of them and smullyan's book has arrived. what is the difference between two books? and do you think that can I read your book without difficulty with my background? i don't know anything about recursive something. I just have read irving copi's book. propositional logic, predicate logic – ju so Aug 09 '21 at 10:32
  • 1
    I'd say read mine (goes slowly, really is intended for someone with just your kind of background); and then you will be in a position to appreciate the elegant brevity of Smullyan's wonderful book! – Peter Smith Aug 09 '21 at 10:45