It is possible to come up with arguments, statements and predicates that are self-contradictory or ill defined, such as the riddle about “who shaves the barber?” (google it) and the corresponding attack on set theory… Such are neither true nor false, and could be used to create a confusing, false proof.

This is not specifically to do with infinite series, or set theory, as the “barber paradox” illustrates. This particular proof about prime numbers is well known, and you won’t find a sane and educated mathematician to doubt it!

In short, given any finite list of primes, it’s always possible to find a new one. Therefore the list is infinite. This proof goes back to Euclid’s time.

Anthony says “Prove it”

Assume there are a finite number of primes. If you multiply them all together, you will have a number for which every prime is a multiple. Add one to this number, and you will have a new number which no prime divides. Hence this new number is prime. This is a contradiction, since it is not in the finite set of all primes, so the assumption is wrong and there are an infinite number of primes.

Or, in haiku http://xkcd.com/622/

if p.any? { |f| n % f == 0 }

or

if p.any? { |f| (n % f).zero? }

(IMHO using positive checks like the above is also simpler to read and parse by humans).