Date: Dec 12, 2012 11:27 AM Author: fom Subject: Re: Almost infinite 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.