Indeed, if the past is infinite, it must share certain abstract characteristics, such as are embodied in Whitrow's two criteria, by virtue of its infinitude, with any
other infinite sequence.
Another advantage of a direct formula would be the ability to compare values of p (n) for arbitrarily large n's and thus to prove the existence of patterns, such as properties that repeat along an
entire infinite sequence.
Ramanujan posited that this pattern should go on forever, and that similar patterns exist when 5 is replaced by 7 or 11 — there are
infinite sequences of p (n) that are all divisible by 7 or 11, or, as mathematicians say, in which the «moduli» are 7 or 11.
The cause of the infinite causal chain may have behind
it another infinite sequence of cause and effect, and that one likewise, etc, etc, etc, and all of that likewise had a cause with an infinite sequence behind it, etc, and similarly, etc..
In every dimension, Cohn and Elkies showed, there is
an infinite sequence of «auxiliary» functions that can be used to compute upper limits on how dense sphere packings are allowed to be in that dimension.
If they could be embedded in
an infinite sequence of exchangeable random variables, then conditional independence would follow from de Finetti's theorem (14).
Zbyněk Baladrán, Approximation of
infinite sequences, 2015, HD video, 12 min, installation view © State of Concept, 2017.
«
Infinite Sequence» is an exhibition dedicated to the work of Ignasi Aballí, winner of the 2015 Joan Miró Prize.
-- Strong Law of Large Numbers: The sample mean of
any infinite sequence of independent and identically distributed (iid) random variables with mean x converges almost surely to x.