While I understand the classic proof-by-contradiction which is usually given to prove that there exist infinitely many primes, I am wondering whether one could argue instead like this. I understand that there is probably something very clearly wrong that I am missing however I'm not sure exactly what.
(1) Assume there are finitely many primes.
(2) Then there would be a finite number of natural numbers; since every natural number is the product of primes, there wouldn't be enough primes to produce every natural number.
(3) Since we don't accept that there are a finite number of natural numbers, we should not accept that there are an finite number of primes.