Goldbach's conjecture
Several facts about the real number α can be proved constructively. However, based on the different meaning of the words in constructive mathematics, if there is a constructive proof that "α = 0 or α ≠ 0" then this would mean that there is a constructive proof of Goldbach's conjecture (in the former case) or a constructive proof that Goldbach's conjecture is false (in the latter case). Because no such proof is known, the quoted statement must also not have a known constructive proof. However, it is entirely possible that Goldbach's conjecture may have a constructive proof (as we do not know at present whether it does), in which case the quoted statement would have a constructive proof as well, albeit one that is unknown at present. The main practical use of weak counterexamples is to identify the "hardness" of a problem. For example, the counterexample just shown shows that the quoted statement is "at least as hard to prove" as Goldbach's conjecture. Weak counterexamples of this sort are often related to the limited principle of omniscience.