From your first paragraph, I assume that each cycle of an individual universe contains different people, doing different things, from those in preceding and following cycles.
Exactly; a 'Bill S' and a 'finiter' discussing infinity would never have happend before, and will never again happen, though humans 'will be' present in all cycles.
I would maintain that the term 'infinite number of times' is not correct because it implies that infinity/eternity can be measured in terms of time.
You say "the infinity in time cannot be removed completely", as though the addition of the word "completely" means that you have removed enough of it to be able to discount it. (more renormalisation) I would maintain that the infinity must either be in your calculations, or not. You can't do away with part of it and ignore the rest.
My reasoning is also the same I think. Eternity cannot be measured in quanta of time. The 'finite time required for a single cycle of events' is not exactly a constituent of eternity. The time is continuous, not quantized, and always moves forward.
The infinity never comes in the calculations because all calculations are restricted to the finite universe. The 'infinite' part remains in the assumed part of the picture for which there is no logical (mathematical) proof. Science or physics can only explain the 'finite' part, and for this there should always be a logical (mathematical) proof.
In short, in my opinion, our picture of the cosmos will always consist of two parts: the 'assumed' part and the 'logically assumed' part. The former has no explanations regarding why it is so, we can say 'it is so' because 'it is so'. For the latter, we will have an answer for all questions regarding why it is so. The latter part comes under science. However, it cannot exist without the former, which is the foundation. The former part, though assumed and unexplainable, should be in agreement with the latter, and should be as minimum as possible. Then only will our picture of the cosmos be logical.
