If there were only finitely many primes, then, by multiplying them all together and adding 1, you would get a new number that could not be divided by any prime at all, which is impossible. (This new number would leave a remainder of 1 if it was divided by any of the numbers on the supposedly finite list of primes; so it would have to be either a prime number itself or divisible by some prime that was not on the original list. In either case, the original finite list of primes must be incomplete. So no finite list can encompass all the primes.
