The Universal Geometric Set

Russell E. Rierson

A simple[trivial?] postulate that gives a "Universal Set" and resolves the "set of all sets" paradox[in the geometric sense]:

A circle of radius R, is isomorphic to a circle of radius 1/R.

[1/R]<--->[R]

For any arbitrarily large circle of radius R, there is an exact correspondence with a circle of radius 1/R, such, that the product R*[1/R] = 1

matt grime

and this resolves russell's paradox? so where is the set of all sets that do not contain themselves in this construction? and in what sense are you using isomorphism? in what category are your morphisms?

Russell E. Rierson

I don't think it resolves "russell's" paradox without some more work.

All circles are isomorphic to each other because they have the same shape. Likewise, all squares are isomorphic to each other. Now if sets can be transformed into geometric shapes, more specifically, circles, or "geometric shape-equivalents", the largest possible set with a geometric radius R, has a corresponding twin with radius 1/R.

matt grime

well, when you've figured out what it is you're trying to prove let us know.

Russell E. Rierson

Here is a definition of the "Euler characteristic":

http://en.wikipedia.org/wiki/Euler_characteristic


Graph Theory:

http://en.wikipedia.org/wiki/Graph_theory


If a polyhedron has V vertices, F faces, E edges, and is topologically equivalent to the sphere, the equation is:

V + F - E = 2

2 is the "Euler characteristic" of the polyhedron.

Sets that are members of themselves correspond to a geometric form. Sets that are not members of themselves correspond to a different? geometric form.

Interesting.

Gokul43201

I thought circles were isomorphic with squares - they don't have to have the same shape.

And the elements of the set transform into...?

And I thought the paradox involved the cardinality of the Power Set being bigger than the cardinality of the Set (of all sets). I may be wrong...but what does this have to do with isomorphisms ?

Russell E. Rierson

Circles are homeomorphic to squares, not isomorphic...?

http://www.rdrop.com/~half/Creations...riangleShapes/

Elements of a set can be characterized as sets. All sets can be associated to geometric forms...?


Any circle of arbitrarily large radius R, is isomorphic to a circle of radius 1/R.

The magnitude of R corresponds to the cardinality of the powerset.

Is the set of all geometric forms, a geometric form?

Can Venn diagrams correspond to light cone cross sections?

Gokul43201

Can I answer your questions with more questions ?

PS : Yes that should have been homeomorphic. But I'm still not getting the point. What is the resolution of the paradox ?

Russell E. Rierson

Set intersection is a type of multiplication of sets.

The intersection of two circles of radius R and 1/R, respectively:

R*[1/R] = 1

R[<-[->[<-[1/R]->]<-]->]


The "Universal Set"

For the continual expansion of power set circle R, there corresponds circle[infinitesimal?] 1/R.

Hurkyl

Not as far as I know.

Gokul43201

No, it's not ! It's just a process of picking the common elements.

You can have a set A containing millions of even numbers, and a set B containing thousands of odd numbers and you "multiply" them to get a null set ?

Russell E. Rierson

Set intersection obeys the distributive law, which is a multiplicative law:

http://www.jgsee.kmutt.ac.th/exell/Logic/Logic31.htm#13

Two sets without common elements are disjoint.

Russell E. Rierson

Venn diagrams are circles...

Light cone cross sections are circles, ellipses, etc.

Hurkyl

That doesn't mean set intersection has anything to do with arithmetic multiplication.


And what does this have to do with associating all sets to geometric forms?

Russell E. Rierson

Since the circle of radius R is isomorphic to the circle of radius 1/R, the cardinality of Circle with radius R is on the same line[radius] as the infinitesimal 1/R

1/R 0--------0 R

Since they are on the same line, they intersect. But perhaps a new type of set multiplicative identity needs to be derived?





When two light cones intersect, they become "phase entangled". The intersection is much like a "set" intersection.


In ordinary quantum mechanics, configuration space is space itself
{i.e.,to describe the configuration of a particle, location in space
is specified}. In general relativity, there is a more general kind of
configuration space: taken to be the space of 3-metrics {"superspace",
not to be confused with supersymmetric space} in the geometrodynamics formulation. The wavefunctions[Venn diagrams-light cones] will be
functions over the abstract spaces, not space itself-- the
wavefunction defines "space itself".


The resultant 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/forms"
intersect/entangle, and are "in phase", they are at "resonance",
giving what is called the "wave-function collapse" of the Schrodinger
equation. the action principle is a necessary consequence of the
resonance principle.

Hurkyl

Although you're using the words, you don't seem to be doing mathematics, so I'll move this thread over here.

matt grime

no they aren't. what idiot told you that?