Here is one - there are finitely many mathematical symbols (or at least, all mathematical symbols can be defined in terms of a finite core of symbols).

That means the set of all mathematical definitions is countable (i.e. you could assign a whole number to each one, putting them into an infinitely long ordered list).

However, the set of real numbers is uncountable (by Cantor's argument).

Therefore the vast majority of numbers ("almost all" numbers, in a mathematical sense) cannot be defined, even in principle.