Then Euclid's proof does not show that there are infinitely many primes (as is commonly reported).
It is trivial to show that there are infinitely many primes from it though through proof by contradiction as follows:
1. Assume there are finitely many primes.
2. Write a list of all the primes. (Writing a list of finite elements is possible)
3. There exists a prime not in that list (By Euclid's proof)
4. All primes are in the list.
3 and 4 contradict and therefore the single assumption is incorrect. So there are infinitely many primes.
As a sidenote, it seems that proving there are infinitely elements of any set will need to be proven by contradiction. Is there any proof of that?
It is trivial to show that there are infinitely many primes from it though through proof by contradiction as follows:
1. Assume there are finitely many primes. 2. Write a list of all the primes. (Writing a list of finite elements is possible) 3. There exists a prime not in that list (By Euclid's proof) 4. All primes are in the list.
3 and 4 contradict and therefore the single assumption is incorrect. So there are infinitely many primes.
As a sidenote, it seems that proving there are infinitely elements of any set will need to be proven by contradiction. Is there any proof of that?