On 12/11/2012 9:58 PM, 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. > > -drt >
When people look at completeness of the real number system, they are looking at convergent sequences attaining a bound within some finite distance of the origin.
The construction of the real numbers usually involves considering such sequences of rational numbers AS the real number.
The "opposite" of this would be a divergent sequence that is unbounded. The sequence itself never gets to and infinite distance from the origin. But, it grows larger than any convergent sequence.
Hopefully, someone may have a better suggestion for you.