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Topic: Almost infinite
Replies: 19   Last Post: Mar 21, 2013 2:40 PM

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ross.finlayson@gmail.com

Posts: 1,220
Registered: 2/15/09
Re: Almost infinite
Posted: Dec 12, 2012 11:23 PM
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On Dec 12, 8:27 am, fom <fomJ...@nyms.net> wrote:
> On 12/12/2012 2:16 AM, William Elliot wrote:
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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.
>
> As I was not concerned with making a statement
> so precise as to be incomprehensible to someone
> asking a question about naive intuition, I was not
> making a claim about every sequence that does not
> converge.
>
> You know just enough mathematics to demonstrate
> the poverty of your personality.
>
> As I recall, when we first met, you were
> unable to recognize a properly negated conditional
> in a proof.



So, who are you?

It's in the charter for USENET you post with your real name.

Slackers.

I wouldn't say this is any better, but why not frame it in terms of
expectations of discrete events about the origin, where, as the bound
increases to where for a given distribution that the samples are not
distinguishable from those of an unbounded domain, that it is
practically infinite.

Another way to look at it is where, there are basically twice as many
elements as needed to establish the period of a periodic function,
there are enough to reconstruct the function.

http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem

Another notion is that there would be 2^n many, for general rational
expansion.

Basically in terms of sampling with replacement and sampling without
replacement, sampling without replacement there're a lot more copies
to be practically infinite.

For example consider a bag with black stones and white stones. If
there are enough of each then, selecting them sees no discernable
difference from that they are infinite, where of course it is
exponential or so how many that is.

Regards,

Ross Finlayson



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