```Date: Dec 12, 2012 12:57 AM
Author: fom
Subject: Re: Almost infinite

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 thereal number system, they are looking atconvergent sequences attaining a boundwithin some finite distance of theorigin.The construction of the real numbersusually involves considering such sequencesof rational numbers AS the real number.The "opposite" of this would be a divergentsequence that is unbounded.  The sequenceitself never gets to and infinite distancefrom the origin.  But, it grows largerthan any convergent sequence.Hopefully, someone may have a bettersuggestion for you.
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