If we answer false, then there must be an infinite prime number. But infinity is not a number and we have a contradiction. If we answer true, then there must be a greatest prime number. But Euclid proved otherwise and again we have a contradiction. So does the set of all prime numbers contain all finite elements with no greatest element? How is that possible?
Answer
Every natural number is a finite number. Every prime number (in the usual definition) is a natural number. Thus, every prime number is finite. This does not contradict the fact that there are infinitely many primes, just like the fact that every natural number is finite does not contradict the fact that there are infinitely many natural numbers. You can have infinitely many finite things, and there won't ever be a biggest exemplar.
To make things a bit more complicated (and a lot more interesting), there are extensions of the set of natural numbers that do contain infinite numbers, and even infinite prime numbers. For instance, in any hyperreal extension of the reals, there is a system of hypernatural numbers. Some of these hypernatural numbers are finite and some are infinite. The finite ones are just a copy of the usual set of natural numbers and the primes in it are the usual primes. For the infinite hypernatural numbers, there are also prime numbers. For instance, the hypernatural represented by the sequence $(2,3,5,7,11,13,17,19,\cdots )$ is an infinite prime number.
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