9
votes
6answers
1k views

Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity. The wikipedia article on constructive proof begins, "a constructive ...
8
votes
5answers
2k views

Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?

Gödel's original proof of the First Incompleteness theorem relies on Gödel numbering. Now, the use of Gödel numbering relies on the fact that the Fundamental Theorem of Arithmetic is true and thus the ...
6
votes
2answers
898 views

When is a statement provable?

We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc. We can also ...
13
votes
2answers
2k views

Clarification of Gödel's second incompleteness theorem

I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific ...