Why not both true and false? Why can't some mathematical statement (or whatever is the correct term) be both true and false?
For example we can prove (e.g. by induction) that $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive integers $n$. But how can we be sure that no one will ever find a counter example? What if someone claims that $1+2+3+\cdots+1000$ equals (e.g.) 500567 and not 500500, which is what the above formula claims.
Another example: Why is it impossible for someone to come up with three integer $a$, $b$ and $c$, for which $a^3+b^3=c^3$ (contradicting Fermat's Last Theorem)? This bothers me even in the simple intuitive level.
Then I have heard about Gödel's incompleteness theorems, second of which says (at least this is how I have interpreted it) that an axiomatic system cannot prove its own consistency. So doesn't Gödel's second incompleteness theorem say basically that "anything is possible"? ...that there can be an integer $n$ for which $1+2+3+\cdots+n \neq \frac{n(n+1)}{2}$ or that there can be integers $a$, $b$ and $c$ for which $a^3+b^3=c^3$?


For example we can prove (e.g. by induction) that
  $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive integers $n$. But
  how can we be sure that no one will ever find a counter example?

The answer to "how can we be sure that no one will ever find a counter example?" is that we can prove (e.g. by induction) that $1+2+3+\cdots+n=\dfrac{n(n+1)}{2}$ for all positive integers $n$.
P.S. "Counterexamples" of the kind whose existence is suggested by Gödel's incompleteness theorem are not in fact numbers in the sequence $1,2,3,\ldots$, that can be reached after some finite number of steps, like the number $1000$.  Rather they are members of systems of "numbers" that satisfy all of the axioms within some system that admits algorithmic proof-checking.
