TITLE: Are there non-diagonal proofs for Cantor's continuum and Godel's incompletness theorems?
QUESTION [12 upvotes]: There is a formal definition for the notion of a formal proof.
Question 1. Is there any formal definition for the notion of a diagonal formal proof?
Consider the following theorems both proved by diagonal proofs.
Theorem 1. (Cantor) $2^{\aleph_{0}}>\aleph_{0}$
Theorem 2. (Godel) $ZFC\nvdash Con(ZFC)$
Question 2. Are there any known non-diagonal proofs for these theorems?
There are proofs for non-existence of formal proofs for some mathematical statements. Now Assuming an affirmative answer for the question 1,
Question 3. Are there proofs for non-existence of non-diagonal formal proofs for some mathematical statements? Are Cantor and Godel's theorems in this category?

REPLY [4 votes]: About 2 (2nd Gödel) see http://andrescaicedo.files.wordpress.com/2010/11/2ndincompleteness1.pdf, quoted in What are some proofs of Godel's Theorem which are *essentially different* from the original proof? (the zoo of Gödel's proofs).