Date: Dec 11, 2012 10:58 PM
Author: David R Tribble
Subject: Almost infinite

We see the phrase "almost infinite" (or "nearly infinite", or "infinite
for all practical purposes") in much literature for the layman, usually
to describe a vastly large number of combinations or possibilities from
a relatively large number of items. For example, all of the possible
brain states for a human brain (comprising about 3 billion neurons), or
all possible combinations of a million Lego blocks, etc.

Obviously, these are in actuality just large finite numbers; having an
infinite number of permutations of a set of objects would require the
set to be infinite itself, or the number of possible states of each
element would have to be infinite. Most uses of the term "infinite
possibilities" or "almost infinite" are, in fact, just large finite
numbers. All of which are, of course, less than infinity.

But is there some mathematically meaningful definition of "almost
infinite"? If we say that m is a "nearly infinite" number, where
m < omega, but with m having some property that in general makes it
larger than "almost all" finite n?

Personally, I don't think there is such a definition; but then I would
enjoy being proved wrong.