Date: Fri, 30 Jan 1998 01:38:26 +0200
From: Metin Aktay <maktay-AT-superonline.com>
Subject: Re: PLC: set of all sets


> eric.dean wrote:
>  It's related, in my
> > mind, to the way in which "infinity" is defined mathematically.  One
> > formulation is: a set is infinite if it has a proper subset of itself which is
> > the same size as itself -- think of the whole numbers and all of the even
> > numbers.  That's not quite the same thing as infinite in the general sense of
> > unlimitedly large, but it serves lots of mathematical purposes.
> >
>Reg Lilly wrote:
>  
> I know the 19th century mathematician Geoge Cantor proved that some infinite
> sets are more infinite than others.  You have to love the notion of different
> sized infinities.  Sorry I can't do the math for you on that!

I think you are referring to countable and uncountable infinities.

I do not know why the interest but i will be glad to share the little
understanding I have of the matter without, I hopelessly hope, being
obtusely mathematical.

For example the set of natural numbers is infinite and yet countable,
that is a process of counting can be begun, no matter how hopeless, as
1,2,3,4,....

Ingenious ways of such counting processes can be found for many sets
even though they are acceptedly infinite. 

An example in this regard is the set of integer fractional numbers. To
count one creates a matrix with integers denoting both columns and rows,
each point in the matrix being one fractional number made up of the
division of the integers denoting its column and row position. If one
then proceeds to count this matrix, beginning at top left corner, in a
path of crisscross diagonals which get bigger and bigger, one ends up
with a semblance of counting the elements of the set of all integer
fractions.

A contrary example would be the set of all decimal fractions where I do
not know of a method which claims to start on a path of exhaustive
counting with an accepted sense of making progress.

All the above are important concepts in "grokking" the completeness of
the real number line, that is the lack of "hole"s in the real number
line, among other things. The completeness of the real number line is
crucial in our acceptance of the set of real numbers as a representative
all possible numerical values in any real life modeling.

Having said all that, I would like to draw attention to the fact that
all mathematics is a modelling game built on definitions. The concepts
of "set", "element", "number", "infinity", and so on, are all
definitions. And the derived interactions between these concepts are
arrived at through applications of formal logic, which again consists of
predefined rules and operands. 

And a paradox is just an inconsistency in the model devised, nothing
more, nothing less. Such as the existence or non-existence of the
set-of-all-sets, which, when defined with qualifications agrees with the
predefined model, and when taken loosely, does not agree with the
predefined model.

Paradoxes do not exist in the realm of nature but solely in devised
models. And "solving a paradox" is just readjusting some definitions in
the model causing it.
One is always well advised to think of Zeno's Arrow which does hit the
target no matter how paradoxically its path is portrayed.

One final word about the real life corollary to the-set-of-all-sets, the
universe itself. If one thinks of grasping-meaning-of/understanding
something requiring a comparison with something else, then,
the-set-of-all-sets, the universe, escapes understanding by definition
as it leaves nothing out of itself to compare it with, not even the
observer. Thus, unsurprisingly, it is not easy to "grok".

So as much as Russell, Cantor, Goedel, Tarski, and so on, have modeled,
there is still Witgenstein at the corner saying that the unspeakable is
unspeakable. 

Metin Aktay

Businessman from Istanbul, Turkey
Home : Ihsan Aksoy sok, EVA apt. No:7/2, Camlik Etiler
       Istanbul 80600, Turkey
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