> 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.
There are some.
Takes longer to calculate with a supercomputer than you and your kids will live.
A hard problem taking exponential time rather than polynomial time to calculate.
y is much greater than x, x is much smaller than y, x << y, when for all n in Z, nx <= y.
Faster than the speed of light. Nearly light speed, ie 180,000 miles/second or faster.