It seems you are missing a key hypothesis in the statement of Gödel's incompleteness theorem. You wrote in a comment:
Godels theorem is proved In a certain system L and shows that for any F (strong enough to do arithematic) P is true and unprovable.
This is incorrect. Gödel's theorem is proved in a certain system $L$ and says the following. Suppose $F$ is a formal system which is strong enough to do arithmetic and consistent (that is, it does not prove a contradiction). Then a certain arithmetic statement $P$ (which is defined in terms of the system $F$) is true but is not provable in $F$.
Now you want to substitute $L$ for $F$. That's fine, but in order to apply Gödel's theorem, you need to know that the hypotheses are satisfied. That is, you need to know that $L$ is strong enough to do arithmetic and that $L$ is consistent. Verifying the first hypothesis is easy, but verifying the second hypothesis is not easy at all. Before you can carry out your argument, you need to prove (within the system $L$) that $L$ is consistent.
In fact, your argument does work (modulo some details that are technical but important) if $L$ could prove that $L$ is consistent, and would reach a contradiction. So actually, your argument shows that if $L$ can prove that $L$ is consistent, there is a contradiction in $L$ (and so $L$ is not actually consistent at all!). This is exactly the statement of Gödel's second incompleteness theorem, which says that no consistent formal system strong enough to do arithmetic can prove its own consistency.
(The technical details: to reach a contradiction, you need to not prove $P$ within $L$, but prove that $L$ proves $P$ within $L$. That is, $L$ needs know not just that $P$ is true, but that $L$ can prove $P$, since this is what contradicts the fact that $P$ is unprovable in $L$. The fact that if $L$ proves $P$ then $L$ proves that it proves $P$ is a consequence of being able to do arithmetic in $L$.
These details are important because in order to prove that $L$ proves $P$, you have to actually have an honest proof of $P$ within $L$. If you have a proof within $L$ that $L$ is consistent, then you get such an honest proof of $P$ from Gödel. But if you just assume for a contradiction that $L$ is consistent, you don't get such a proof and so you cannot conclude that $L$ proves $P$; instead you only know that $P$ is true. So you can't reach a contradiction: there is no contradiction between the two statements "$P$ is true" and "$L$ cannot prove $P$".)