I was actually wrong about this!!! After reading your comment, I thought about it more. We can create an infinite number of integers by exponentiating a single prime, like 2, an arbitrary number of times. So we could do 2^2, 2^(2^2), 2^(2^(2^2))), etc. Of course, we can easily find numbers, like 3, that can’t be created in this manner. But without a proof, it’s not immediately obvious that we couldn’t create all the integers by multiplying a finite set of primes an arbitrary number of times. Even if we’ve proven than every integer greater than 1 has a unique prime factorization, we need a separate proof to show that infinite primes are needed to construct all integers greater than one, rather than just one more prime than we currently have in the set. This is the proof that Euclid’s Theorem provides.
I was actually wrong about this!!! After reading your comment, I thought about it more. We can create an infinite number of integers by exponentiating a single prime, like 2, an arbitrary number of times. So we could do 2^2, 2^(2^2), 2^(2^(2^2))), etc. Of course, we can easily find numbers, like 3, that can’t be created in this manner. But without a proof, it’s not immediately obvious that we couldn’t create all the integers by multiplying a finite set of primes an arbitrary number of times. Even if we’ve proven than every integer greater than 1 has a unique prime factorization, we need a separate proof to show that infinite primes are needed to construct all integers greater than one, rather than just one more prime than we currently have in the set. This is the proof that Euclid’s Theorem provides.