To show that the existence of an actually existing infinite set leads to contradiction, assume the existence of an infinite set of brutely distinguishable points. Now another point pops into existence. The former and latter sets are indistinguishable, yet they aren’t identical. The proviso that the points themselves are indistinguishable allows the sets to be different yet indistinguishable when they’re infinite, proving they can’t be infinite.

Suppose I assigned each of the points different spacetime coordinates. Now another point pops into existence at a different spacetime coordinate. The two sets of points are distinguishable because one of them has a point at a spacetime coordinate that the other one doesn't. I don't see the contradiction here.

But if infinite quantities exist, then relative frequency should equal probability.

Um. Why?

In general, I agree with the other comments that this post is unclear and not well-written. In particular, I agree with shminux's comment that your definition of "exist" is unclear. You should taboo it.