The number of theorems that can be proved is infinite, but each proof is finite.
If assuming that there is a longest proof leads to a contradiction, then there is no longest one. That doesn't imply that any of them is infinite; by analogy, no prime number is infinite.
Yes, quite right there, VA. I still think that one can show that the cardinality (if I can use that term) of some theorem must be "Aleph nought," the cardinality of the positive integers. Let me think about it some more.