The Martian and the Earthman - and the set concept

1. Apr 10, 2004

Organic

Earthman: x stands for an actual thing; therefore x cannot be used together with {} if we mean that {} is the notation of the empty set. Shortly speaking, {} is our notation of the empty set and {x} is a general notation for a non-empty set.

Martian:
x stands for general notation of any concept, for example: x=something, x=nothing.
Therefore {x} is our general notation for both empty and non-empty sets.

Earthman:
|{}|=0, |{{}}|=1 and there is no middle state.

Martian: I agree that there is no middle state; therefore we can say that there is a phase transition from {} to {{}} and vise versa.

Earthman: In a non-empty set there can be finitely many or infinitely many elements.

Martian: In a non-empty set there can be finitely many or infinitely many elements, but there is another content which is the opposite of emptiness. We call it fullness and it is notated by {__}, which means that no finitely or infinitely many elements can be found in it. Shortly speaking, it is a one and only one element that cannot be constructed by any collection of points or segments, and |{__}| = 1.

Earthman: What to you mean by no segments. For example: I can take any ___ and find its length by using finitely or infinitely many sub-segments.

Martian: Yes, but in this case you don’t have a one solid ____ but a collection of finitely or infinitely many elements. Shortly speaking, I am not talking about length measurement, but on the structural difference between a "one piece" element, which is not a point, and "many pieces" element.

For example: No broken glass is an unbroken glass.

Earthman: So how can you take ___ as something that can be used by Math Language?

Martian: As I said before: x stands for general notation of any concept, for example: x=something, x=nothing.

Therefore {x} is our general notation for both empty and non-empty sets.

When x is emptiness, we are using it as the unreachable weak limit of Math Language.

When x is fullness, we are using it as the unreachable strong limit of Math Language.

Both emptiness and fullness cannot be used as available inputs for any mathematical system, but they give us the lowest and the highest limits of Math Language.

Shortly speaking, any possible mathematical system can be found between these limits.

Earthman: I understand that emptiness cannot be used as an input, but why fullness cannot be used as an input? After all we have here an existing element.

Martian: You can use fullness only if you break it to pieces of information, but then you have no fullness but finite or infinitely many pieces, and as I explained before, no finite or infinitely many pieces can be in fullness state.

Shortly speaking, |{}|=0, |{a}|=1 and there is no middle state between them.

The same holds for |{a}|=1, |{__}|=1 but as you see, this time the cardinality is the same but {a} is one of many where {__} is The unbroken one, and there is no middle state between them.