I thought I understood Gödel's Incompleteness Theorem to say:
Starting from ZF, there only a countable number of proofs you can write
The number of possible conjectures is uncountable.
Thus, there is a conjecture S such that you can't write down a proof of S, but you also can't write down a proof of -S.
However, I later realized the number of conjectures you can write down is also countable.
What am I missing?