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I think the point is you can easily remove the "deficiencies" of Euclid's proof. Basically, say: let p_1, ..., p_n be the first n primes. We can construct a new
by obastani 11y ago
I think the point is you can easily remove the "deficiencies" of Euclid's proof. Basically, say: let p_1, ..., p_n be the first n primes. We can construct a new prime by examining p_1...p_n + 1. This process can be continued indefinitely.
In fact, I think the statement "this process can be continued indefinitely" is quite misleading -- to formalize this, I think you need proof by contradiction. So, the difference is the syntax used, not the actual semantics of the proof.
- IngoBlechschmid 11y agoThere 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.