There are countably many proofs, but a proof does not entail a single truth. Many truths are entailed by a single proof, in fact an uncountable number of truths

There are countably many proofs, but a proof does not entail a single truth. Many truths are entailed by a single proof, in fact an uncountable number of truths can be entailed by a single proof. That does not mean that all truths can be written down or enumerated, but strictly speaking it is not sufficient to conclude that due to the fact that the set of all proofs are countable, that the set of all truths entailed by the proofs must also be countable. None of this should be taken to violate Godel's incompleteness theorems. Finally, it's worth mentioning that not all formal systems are limited to finite proofs. There are formal systems where theorems as well as proofs can be countably infinite in length and where there are uncountably many proofs. These systems, known as infinitary logic, are often reduceable to second order logic and hence are incomplete.