What is 'the least number which cannot be described in less than nineteen syllables'? Is this not a description of it, only 18 syllables long?
What about an inconsistent set of axioms? The set of all propositions is a set of axioms, so there you have it already, though a set of axioms technically does not prove anything, as axioms are just propositions. You need inference rules in order to prove anything.The problem appears to be that, for any language sufficient enough to describe all the propositions we'd want to create, there does not exist a set of axioms which could prove every proposition (Gödel's incompleteness theorem).