TITLE: Is no proof based on "tertium non datur" sufficient any more after Gödel?
QUESTION [11 upvotes]: There are many proofs based on a "tertium non datur"-approach (e.g. prove that there exist two irrational numbers a and b such that a^b is rational).
But according to Gödel's First Incompleteness Theorem, where he provides a constructive example of a contingent proposition, which is neither deductively (syntactically) true nor false, we know that there can be a tertium.
My question: Are all proofs that are based on that principle useless since now we know that a tertium can exist?

REPLY [3 votes]: I think that many of the responses here are overlooking the key part of your question, "Are all proofs that are based on that principle useless?".  The question of whether LEM is not merely formally consistent but actually useful, i.e., pertaining to reality, is a deep question that has been debated for at least a century (since Brouwer).  One partial answer to your question is that no, not all such proofs are useless, because often they involve LEM on propositions which are indeed decidable.  In constructive mathematics, $\varphi \vee \neg \varphi$ is precisely what it means for a proposition $\varphi$ to be decidable, and making use of this fact in a proof amounts to calling a decision procedure.