Let us consider the following two theorems by Gödel:
1) Any consistent formal system $F$ within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of $F$ which can neither be proved nor disproved in $F$;
2) For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in $F$ itself.
Like others, probability theory is also an axiom-based system with 3 axioms and supports minimum arithmetic.
From the above information, it looks like that probability theory also suffers from Gödel's incompleteness theorem. Can't it? If not, then where I am ging wrong?
Thus is there a possibility that there may exist two statements in probability theory that contradict each other and existence of unprovable facts?