to explain truth, tarski created semantic theory.
as for godel's first incompleteness theorem, 定理 mentioned a few times here. this theorem was significant because it shows that there is no consistent, complete system of arithmetic. also turing (as well as church) showed that no consistent system of arithmetic is decidable, that is, we can not find an effective way to decide if a statement is true. these two theorems put hilbert's question to rest.
being a philosopher, you must have heard of russell's paradox and his co-authored principia mathematica.

haha