I have heard of (still not learnt) Gödel's incompleteness theorem which says that there are some statements unprovable.
Now suppose we suspect that there is some rule. And the rule remains unproven despite several decades', or several centuries' effort (just as the conjectures). Then we suspect a possibility that actually the rule itself is unprovable.
So, is there some trivial method to determine whether a statement is provable or not? I guess there is, for scientists have marked some statements as axioms and have left them unproven.
If this question is not too stupid to be answered, and any of you like to answer it, please provide an example where the unprovability of a well-known axiom is proved (preferably, simple geometric axiom such as playfair's axiom).