Consider that if F() is uncomputable, it seems extremely dubious that humans have some mechanism that lets them create Turing machines that produce arbitrarily-large values of F.
I'm probably wrong on the strict technical merits but those technical grounds are unduly Platonist for my taste.
2. Is trial division the Sieve of Eratosthenes? Maybe it's the same algorithm, just a particularly bad implementation!
I'm with R. A. Fisher on this one. If the idea was just some way to laboriously generate primes, it wouldn't be famous, even among Eratosthenes' contemporaries. The whole thing that makes the sieve notable is that you can apply arithmetic sequences (which translate into straight lines on a grid!) to cut down your work.
The counterargument seems to be that Eratosthenes might well have performed "skip, skip, skip, skip, mark" instead of jumping ahead by five.