# On the set concept

1. Apr 19, 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.

2. Apr 19, 2004

### Russell E. Rierson

It is clearly self evident, that, if x = something, then x cannot equal ...nothing.

Let x = something

Let not-x = nothing

[ x or not-x] is a tautology

Let [x or not-x] = T(x)

If T(x) then A

If T(not-x) then B

[A or B] = T(T(x))

etc...

A set is a quantity that has an identity that distributes over its elements.

The set of all chickens is predicated by the "chickeness" of its members.

C[a,b,c,...,n] = [Ca, Cb, Cc,...Cn]

The set becomes a nonparadoxical member of itself due to its distributive identity, an informational construct called abstract containment. The abstract contains the concrete as the concrete contains the abstract.

The identity relates to all members of the set. Einstein said that space and time become modes by which we think, not conditions in which we live, if memory serves.

DNA is an algorithm, a finite set of instructions, which can construct a carbon based life form.

The life form physically contains the DNA and the DNA contains the life form in an "abstract" sense.

At a fundamental level of existence, it is postulated that "nature" could be constructed of tiny strings, and those strings, loops, or branes, could even be constructed of string "bits".

These bits could encode information, analogous to the universe's "DNA"? A set of instructions built into the fabric of space/time and mass/energy?

At the most fundamental length scales, the fundamental paticles, called "strings", could be constructed of even more basic units i.e. bits? analogous to a computer code?

1010100010...etc.

Universal algorithms?

This assumption seems to hint for a designed universe, or even stranger still, a universe that is a type of life form...???

Some interesting ideas on "string bits":

http://xxx.lanl.gov/PS_cache/hep-th/pdf/9607/9607183.pdf

http://xxx.lanl.gov/PS_cache/hep-th/pdf/9707/9707048.pdf

According to string theory, the uncertainty in position is given by:

Dx < h/Dp + C*Dp

Which points towards a type of "discrete" spacetime?

So a metric space with distance function r(x,y) involes the real numbers R, allowing the metric space to be embedded in the full structure of manifold, M.

We humans need a quantum field theory of curved spacetime, where the interlocking, discrete, yet causally connected structure of spacetime can be described as a manifold, M, with a metric tensor g_ab. Ergo, the difficulty of formulating a classical background metric is taken care of by the background independence of GR.

Of course the breakdown of the "theory" occurs when spacetime curvature approaches the "Planck" scale. Thus the exact criteria necessary for a valid quantum field theory of curved spacetime requires a quantum theory of gravity. Some potential candidates are loop quantum gravity and string theory...

It appears that a quantum theory of a field system differs from a quantum theory of a particulate system due to the fact that a field system has infinitely many degrees of freedom. Yet, for a system with finitely many degrees of freedom, the kinematic structure of the spacetime in question can be determined by the canonical commutation relations for the position and momentum operators. So the "square matrix" hermitian operators determine up to unitary equivalence the position and momentum observables,

...BUT, it appears there could be problems with formulating a spacetime structure that has infinitely many degrees of freedom.

In general curved spacetime, there does not appear to be any preferred notion of "particles". So for a noncompact space, where the natural notions of particles are available in the asymptotic past and the asymptotic future, the canonical commutation relations corresponding to the two, will, be generally unitarily inequivalent, analogously to the phenomenon of the
ultraviolet catastrophie of quantum electrodynamics.

So if my interpretation is correct, it is necessary to take an algebraic approach, which allows one to consider all states arising in all of the unitarily inequivalent Hilbert space constructions on an equal basis.

So spacetime is locally Euclidean[flat] which allows a universal representation from flat spacetime to curved spacetime, the necessary elements of Poincare symmetry to logically define the canonical commutation relations.

Getting back to the question, "What is a set?", we realize that sets are by definition general entities that correspond to a non-condradictory state of affairs, allowing for meaningful interpretation.

Last edited by a moderator: Apr 20, 2017
3. Apr 19, 2004

### WWW

Maybe x is like a wave function of probabilities in QM?

4. Apr 19, 2004

### Russell E. Rierson

Excellent point WWW.

A topological space is a set X along with a happy family of subsets of X, called the open sets, requred to satisfy certain conditions, like the empty set and X itself are both open, if the subsets of X, U and V are open, so is the intersection of U and V, of course! And if the sets U_a of X are open, then so is the union of U_a. The collection of sets taken to be open is called the topology of X. An open set containing a point x, which is an element of X, is called a neighborhood of x. The complement of an open set is called "closed".

If M is an m-dimensional manifold and N is an n-dimensional manifold, then M x N is an (m+n) dimensional manifold.

So simultanaety "S" is a spacelike hypersurface or "slice" through spacetime that cuts through event P, with a set of observers having worldlines crossing the simultanaety "simultaneously-orthogonally" having clocks that all read the same "proper" time at the instant of crossing.

The metric spaces are thus defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system. When the "wave-functions" intersect, and are "in phase", they are at "resonance", giving what is called the "wave-function collapse" of the Schrodinger equation.

So if x is categorized as a general entity, capable of being both something and nothing, it creates a slight problem, in that a general entity is of itself NOT nothing...

5. Apr 19, 2004

### WWW

So, I think that the Martian logic is the probability of opposite states (notated as {x}) before we get one and only one of them as the value of x, which is nothing XOR something, in this case.

6. Apr 20, 2004

### Russell E. Rierson

First the term "nothing" must be rigorously defined. What is "nothing"?

Nothingness must be approached as a limit. A removal of "somethings", until all that is left is ...nothing.

I prefer to see "nothing" as an infinite symmetry. The Universe and time, is then a sequence of "symmetry breaking".

7. Apr 20, 2004