TITLE: Gödel says: countable proofs, uncountable conjectures?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: While I agree with the other posters that the direct connection you're looking for doesn't exist, there's another proof of Gödel's theorem that probably has much the flavor you're after; in essence, some statements are unprovable because they're too complicated!  In particular, consider the statement '$x$ is the smallest number that cannot be computed by a (particular universal turing machine) program of length less than $N$' (where $N$ is a number we'll give specifically later).  Since there are only $2^N$ programs of length less than $N$, such an $x$ must exist, whatever we take $N$ to be.
But the description of this statement takes $k_1\log N+k_2$ bits for some constants $k_1$ and $k_2$ (since $N$ can be written in binary); and the axioms of the formal logic system, along with a procedure to enumerate all consequences of those axioms, takes only a finite number $k_3$ of additional bits.  So if the statement '$x$ is the smallest number that cannot be computed by a program of length less than $N$' were provable, then its proof — as enumerated by the Turing machine — provides a program of length $k_1\log N+k_2+k_3$ bits for computing $x$.  But there's surely an $N$ such that $N\gt k_1\log N+k_2+k_3$, whatever we take $k_1$, $k_2$ and $k_3$ to be, so if this statement were provable then we'd have a contradiction - a description for $x$ of length less than $N$!  
This means that for large enough $N$ the statement must be unprovable for all $x$, but it's surely true (as noted initially) for some $x$.  In essence, the axioms of the formal system only hold a finite amount of information in them, and so statements much more complicated than that finite amount of information are inherently unprovable.  (This interpretation of the argument — which is essentially a formalized version of the Berry paradox — is due, AFAIK, to Gregory Chaitin, and you can check his writings for quite a bit more information on it, pun intended.)