Can |R| be uncountable but not a Continuum?

[Doron Shadmi]

Hi selfAdjoint,

In general, through the structural point of view, we have two levels
of XOR retio:

Level A is: ({} XOR {.}) or ({} XOR {_})

Level B is: {.} XOR {_}

3) A distance somehow would involve a partial state where the set for a number would in one view be empty but in another view have members, whic is a contradiction.

Does this represent your idea?

No it doas not. Through my point of view there are 3 structural types of sets:

{}, {.}, {__}

In Common Math there are only 2 set's types: {}, {.}


The 3 types:
-----------

{} = The Emptiness = 0 = Content does not exist.

Let power 0 be the simplest level of existence of some set's content.

{__} = The Continuum = An indivisible element = 0^0 = Content exists (from the structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, But because it exists (unlike the emptiness) its full notation = 0^0).

(You can break an infinitely long continuum infinitely many times, but always you will find an invariant structural state of {.___.} which is a connector between any two break points.)

{...} = The Discreteness = Infinitely many elements = [oo]^0 = Content exists.

Any transformation from {} to {__} or {...} is based on phase transition, because we have 0(=does not exist) to 1(=exists) transition.

So, from a structural point of view, we have a quantum-like leap.


Now, let us explore the two basic structural types that exist.

0^0 = [oo]^0 = 1 and we can see that we can't distinguish between
Continuum and Discreteness by their Quantity property.

But by their Structural property {__} ~= {...} .

From the above we can learn that the Structure concept have
more information than the Quantity concept in Math language.