No theoretical system can survive without being aware to its limitations.
It means that any
x output can be only a model(
X ) input.
Shortly speaking,
x =model(
X ).
Math is first of all a form of theory, therefore any concept that can be used by it is only a model(
CONCEPT ).
For example, let us take infinity concept.
If
INF is infinity itself (= actual infinity) , then
inf =model(
INF )=potential infinity.
Please look at this model for better understanding:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf
In this way we first of all aware to our input limitations, which are:
No input = model(
EMPTINESS ) = lowest limit.
No input = model(
FULLNESS ) = highest limit.
If we translate this to set's representation then:
{} content = model(
EMPTINESS ) = lowest limit.
{__} content = model(
FULLNESS ) = highest limit.
Between these limits ({},{__}) we can find
inf =model(
INF )=potential infinity, where
inf has two input forms:
{.} = singleton, which is a localized element.
{.__.} = non-singleton, which is a non-localized element (connect at least two different singletons).
{.} and {._.} can appear in two basic collections:
Collection {
a ,
b ,
c } is finitely many elements.
Collection {
a ,
b ,
c , ...} is infinitely many elements (=
inf ) .
Any non-empty collection which is not a singleton, is an association between {.} and {._.}, for example:
b b
{a , a}
. .
| |
|___|_
|
{a , b}
. .
| |
|___|
|
I opened a new thead for this at:
https://www.physicsforums.com/showthread.php?s=&threadid=14416