[P.O.E.]

The proof is, take all the primes you know of, multiply them together, and add 1. The new number can't possibly be wholly divided by any of its divisors -

Not sure I'm following you here. E.g., suppose you only "knew about" the primes 3 and 5. Multiplying them and adding 1 gives you 16, which is certainly dividable. (3x5x7)+1 is 106, etc.

[coloradan]

Yes, 16 is dividable - but not by 3 nor 5. What I wrote was, the new number is either prime, or it's a composite number containing prime factors not on your original list. The latter is the case here - 2 wasn't on your list. Let's do this again, now knowing of 2, 3, and 5. Their product is 30. Adding one makes 31. In this case, it's a new prime. You will never, ever run out: this process will always generate new primes, or composite numbers with all-new prime factors. On the other hand, this process doesn't guarantee it will generate a complete set - only that the set is infinite.

[coloradan]

I forgot to mention: 106 is 2 x 53 - two new primes not on your list of 3, 5 and 7.