“If you don’t believe Euclid’s proof that there are infinitely many primes, then I don’t know why seeing a formal proof in ZF set theory should quell your doubts. You could deny Modus Ponens, you could demand a formal proof for the assertion “The infinitude of primes is a theorem of ZF” itself (ad infinitum), you could refuse to identify the formal notions of ‘prime’ or ‘infinity’ with the informal ones, and so on.