Infinite cannot imply exhaustive as it would negate the whole concept of infinite (boundless). Exhaustive only goes so far as explanation for what a computer by us can generate for a step-by-step sequence, for primes. Thus, it has its

**human** limits. That doesn't mean there's 'limits' to the system of logic on which this numbers game is based. I would disagree with the word exhaustive. I am getting what you

*mean* from a purely syntax model of logic, but, in this scenario, of primes and infinite, it is meaningless. Being or not being exhaustive. That's only relevant in terms of 'if humans wanted to generate all prime numbers'. I think the key question is: 'if humans CAN have the ability to generate all prime numbers'. I.e., the key to the lock.

Why would we want to find them all? Why would that logically and realistically be a goal of ours in the first place? To find all of them -

*literally*? All prime numbers? I don't understand this premise.

It is only to find the system by which primes work. To find order in this apparant chaos that is such an anomaly in the field of such rigorous deductive logic that

*is* mathematics.

Now, if we want to apply mathematical limits to infinite in the treatment of primes, here's a read off Wiki:

Prime number theorem - Wikipedia, the free encyclopedia