On Tue, 11 Dec 2012, fom wrote: > 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. > > 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. > ... sequences attaining a bound within some arbitrary distance of the limit.
> The construction of the real numbers > usually involves considering such sequences > of rational numbers AS the real number.
... involves considering equivalence classes of such sequences of rational numbers as the real numbers.
> The "opposite" of this would be a divergent > sequence that is unbounded.
The opposite of a convergent sequence, by definition, is a divergence sequence. A divergent sequence can approach oo, -oo or oscillates or oscillates within a region.
> The sequence itself never gets to and infinite distance from the origin. > But, it grows larger than any convergent sequence. > The divergent sequence (sin n)_n doesn't grow larger than the convergence sequence (3 - 1/n)_n.
> Hopefully, someone may have a better suggestion for you.