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 $a^3+b^3=c^3$?

It actually might be both true and false at the same time in a way. For example
lim(x->0) f(x)/g(x) 
with f(0)=0 and g(0)=0 might be true or might be false depending on the speed f(x) and g(x) reach the zwero when x reaches zero. You might even get two results depending on the direction from which you reach the 0.
The computer algebra system maxima will return "undecided" in this case.
In other cases
solve(f(x)=0,x);
will return a list of results. This, too is an example where a question can neither be answered by "true" nor by "false".
