Am I correct in saying that a hyper-finite frequentist account of probability would require not just absolute convergence of relative frequencies in the absolute limit but an absolute arithmetic convergence of the actual frequencies (so if the events are equally probable there must only be equal relative frequencies but precisely the same number of each event!;
So that necessarily on even odds, an agent will break precisely if the probability or relative frequencies are precisely 0.5; I say this because I would presume that even only tiny difference say finite difference between the number of A events in ~A events in an infinite collective would disturb the relative frequency if takes into account infinetismals.
It would make 0.5 + infinitesimal and if frequentist require necessary convergence to the probability values, and the probability values is precisely 0.5, then convergence has in some sense failed.
Would this strong kind of convergence have implications on place selection rules and the kinds of dependency that occur between events (events would appears to be more inter-connected, or teleological in a hyperfinite account as it cannot even account for the minutest of differences).
Likewise I presume the laws of large numbers (outside of frequentism would have to be different) ; i presume that even if one says that convergence only occurs with probability 1-infinismal, if infinitesimals infect the frequencies as well I presume that probability of gettting precisely that relative frequency is infinitesimal not 1- infinitesimal and so that infinesimals would have to placed around both the relative frequencies and probabilities values ie
Pr(lim>inf{ relative frequency of A in [Pr(A)-infinesimal, Pr(A) + infinesimal] }=1- infinitesimal; otherwise they would have to say that convernce is almost surely not going to occur, and if they do the above the difference between measure theoretical or more kolmogorovian nonstandard models and nonstandard frequentist models becomes much greater
Likewise could a nonstandard frequentist account for irrational probability values given that the sequences is treated in a finite fashion--- if its still considered countable infinite (whatever is that is analogous to in nonstandard analysis) I presume one could not fit all given real number relative frequency value into the collective as there are only countably many positions; 1.one cannot simply re-arrange as in the standard analysis case, so that becomes a different sequences, 2 perhaps treat two numbers very close together as the same number
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