On 12/12/2012 2:16 AM, William Elliot wrote: > 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.