David R Tribble wrote: > 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.
The key here is what we mean by infinite, of course.
The clearest way to think about the smallest infinity that I have found is to imagine a one-ended chain, loop-free, etc -- literally infinite, not-ended in one direction.
I think that would mean a finite-but-almost-infinite chain would have slightly more than one end, and that would imply an integer between one and two.
In physics, properties are often calculated from the most important effects and then corrections are made from less and less important effects. The orbit of the Uranus can be calculated to a good degree of accuracy with a straightforward application of Newton's laws, but it took the assumption of the existence of another major planet (Neptune, later found) to account for some of the difference between prediction and observation.
If Neptune were infinitely far from Uranus, then its gravitational effect could be ignored. Since its gravitational effect can be *almost* ignored, I suppose that one could say that Neptune is *almost* infinitely far from Uranus.