Quote by fibonacci235
I believe that Fibonacci primes are infinite. Currently there is no proof that there is an infinite number of Fibonacci primes. I was wondering why we couldn't compare the set of Fibonacci primes to the set of Natural Numbers and demonstrate that both have cardinality aleph null? Indeed, why couldn't we do that for Mersenne primes, Sophie Germain primes, Wilson primes etc.

If we show that both have cardinality [itex]\aleph_0[/itex], then we would indeed have shown that there are an infinite number of them. But it's as difficult to prove.
Wouldn't that imply that these sets of prime numbers are infinite? I'm assuming if it were that easy someone who have demonstrated that by now. Clearly, that is not the case, so I am wondering why it is so difficult to prove that these sets of Prime numbers are infinite. After all, Euclid demonstrated that the set of primes is infinite; wouldn't that imply that its subsets would be infinite too?

Not at all. For example {57} is a subset of the primes, but it isn't infinite. Or the "even prime numbers" are also a subset, but this set is finite.