There a ways to formulate "the infinity of the prime numbers" without any recourse to a previously-constructed set of all prime numbers (or even all natural num

There a ways to formulate "the infinity of the prime numbers" without any recourse to a previously-constructed set of all prime numbers (or even all natural numbers). Two such ways are: 1. "For any natural number n, there is a prime number greater than n." 2. "For any natural number m, and for any list of m prime numbers p_1, ..., p_m, there exists a prime number not on that list." Both statements have a perfectly fine constructive proof. The second statement is more or less precisely what Euclid proved. The first statement follows from the second one with a bit more work.