Does every theorem have a short proof? This question is somehow based on my belief that every theorem has a short and simple proof. By "proof" I mean:


*

*Proving an statement

*Disproving a statement

*Proving that a statement is undecidable


Once we have formalized what we understand for a "step" in a proof, could it be proven that every theorem has a proof consisting of less than $n$ steps? If so:


*

*What would be the (minimal) value of $n$?

*Such a proof would be about all proofs so what would it say about itself?

*Could there be (in some sense) proofs with a non-integer number of steps?


surely a theorem doesn't need to conform to any real quality requirements, so it could contain an huge number of steps that need to be proved, greater than any n you think of.  
