Proving God Existence

[Zarith]

The definition of S1 is to me ambiguous. Either you are saying that:
1) S1 contains all states which are separated in time from 1/1/2000 00:00:00 by an amount that is smaller than some arbitrarily chosen fixed constant number T1, or ...
2) S1 contains all states which are separated in time from 1/1/2000 00:00:00 by some finite number, this number being different for different members of S1

If you mean definition (1), then S1 contains a finite number of elements and S2 contains an infinite number of elements.
If you mean definition (2), then S1 contains an infinite number of elements and S2 is empty.

To see why (2) entails an infinite S1, note that given any member of S1 at time T0, a new member of S1 can be generated at T1=T0+1. T0 is finite therefore T1 is too.

This should have been obvious, since you can't start with a set containing an infinite number of elements (the integers), split it, and end up with a finite number of elements in the union of the 2 resulting sets.

The point is that we assumed that infinite number of elements can exist, so we can define (hypothetically) a set with infinite number of elements

If we start by an infinite number of events, then splitting it can either result in 2 infinite sets or one finite and the other is infinite.

then prove that an infinite number of elements is impossible.

It seems there is some confusion on your part about what I said.

Your argument is that splitting a set S (which contains an infinite number of elements) into 2 sets S0 and S1 results in S0 having a finite number of elements and S1 having a finite number of elements. Stay in school?