Is there a proof of Gödel's Incompleteness Theorem without self-referential statements? For the proof of Gödel's Incompleteness Theorem, most versions of proof use basically self-referential statements. 
My question is, what if one argues that Gödel's Incompleteness Theorem only matters when a formula makes self-reference possible?
Is there any proof of Incompleteness Theorem that does not rely on self-referential statements?

There is a weaker version of the first incompleteness theorem that is an almost trivial consequence of an insight from computability theory, namely of the result that 
  there exists a computably enumerable set that is not computable   (*).  

Consequence: the set of true first-order sentences (i.e. true about the 'real' natural number sequence) is not axiomatizable by a c.e. axiom set.
Unfortunately, most proofs of ($*$) have themselves a scent of self-referentiality hanging around them. However, you may want to check out 'simple sets'. Simple sets are c.e. and not recursive, and the standard textbook-proof of their existence is, to the best of my knowledge, the argument that comes closest to a non-selfreferential argument for ($*$).
