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Re: Almost infinite
Posted:
Dec 12, 2012 3:16 AM
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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.
How so? It seems unrelated to his puzzling.
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