Is there a set which cannot be proven to be finite or infinite? The set must be not be proven to be infinite of any cardinality.
I am a novice at set theory regarding these topics, so if this is something that I can solve for myself if I study the right topics, pointing out the topics will be very helpful to me.
Consider the set of all primes $p$ such that $p+2$ is also prime. Then, proving that this set is infinite or finite means proving or disproving, respectively, the Twin Prime Conjecture.