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.