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From: Osher Doctorow on 14 Jul 2010 00:48
From Osher Doctorow

Since Expansion vs Repulsion in all Interaction has a "transition" at
P(A) = 1/2, and P(A) = 1 corresponds in general to "infinity" (whether
reals or integers) while P(A) = 0 is in a sense an "opposite" of
infinity, numbers like 1/2 which are strictly between 0 and 1 in
probability seem to be capable of interpretation as a "part" of
infinity that is not trivial, which makes it arguably roughly between
the cardinality or "number" of integers and the cardinality or
"number" of real numbers.

See Wikipedia's "Continuum Hypothesis" online. Note that both Kurt
Godel (a Platonist) and Cohen (a Formalist) disagreed with the
Continuum Hypthesis, which claims that there is no set with
cardinality strictly between that of the integers and that of the real
numbers. They were two of the greatest Logicians of all time.

What is especially interesting is that 1/2 appears to be the
transition point between Repulsion and Attraction in Interactions.
With discrete random variables or discrete variables like tossing a
coin to see if it lands on heads or tails, a "fair" coin has
probability 1/2 of landing on either side, which does not look like
either infinity or a "partial" infinity, but with continuous random
variables like (random) distance or radius or scaling/scale factor a
or R of the Universe, on a scale of 0 to 1 (1 being infinity), 1/2
would describe "something" halfway between 0 and infinity.

Intuitively, if there is a real "Semi-Infinity" as for example
indicated by 1/2, then although probabilities less than 1/2 of bounded
objects may increase with volumes of the objects similarly to the
Lebesgue integral, there may be some surprises regarding their
unbounded complements and whether they increase with volumes. If
bounded objects A all have probabilities < = 1/2, then unbounded
objects would all have probabilities > = 1/2 as represented by their
complements A ' , but if any bounded object has probability > 1/2,
then its unbounded complement is LESS than 1/2 even though the latter
is "infinite" (being unbounded) while the latter is "finite" (being
bounded). Of course, a reverse "regime" could also be hypothesized.

Osher Doctorow
From: Osher Doctorow on 14 Jul 2010 01:07
From Osher Doctorow

"Continuum hypothesis" keywords bring up 6 papers in arXiv ending in
2008, but "cardinality" brings up 50 (1 in 2010, 7 in 2009),
"cardinality reals" 2, "cardinality integers" 1, "aleph" (symbol for
various infinities) 87 including 5 in 2010 and 2 in 2009, "aleph 1"
20, "aleph 0" 18 (2 in 2010, 2 in 2009).

Osher Doctorow
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