## The Universal Geometric Set

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

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 Recognitions: Homework Help Science Advisor 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?

 Quote by 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?
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.

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## The Universal Geometric Set

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

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

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

 Now if sets can be transformed into geometric shapes,
And the elements of the set transform into...?

 more specifically, circles, or "geometric shape-equivalents", the largest possible set with a geometric radius R, has a corresponding twin with radius 1/R.
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 ?

 Quote by Gokul43201 I thought circles were isomorphic with squares - they don't have to have the same shape.
Circles are homeomorphic to squares, not isomorphic...?

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

 Quote by Goku43021 And the elements of the set transform into...?
Elements of a set can be characterized as sets. All sets can be associated to geometric forms...?

 Quote by Goku43201 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 ?
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?

 Recognitions: Gold Member Science Advisor Staff Emeritus 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 ?

 Quote by 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 ?

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.

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 All sets can be associated to geometric forms...?
Not as far as I know.

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 Quote by Russell E. Rierson Set intersection is a type of multiplication of sets.
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 ?

 Quote by 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 ?
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.

 Quote by Hurkyl Not as far as I know.

Venn diagrams are circles...

Light cone cross sections are circles, ellipses, etc.

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 Set intersection obeys the distributive law, which is a multiplicative law:
That doesn't mean set intersection has anything to do with arithmetic multiplication.

 Venn diagrams are circles... Light cone cross sections are circles, ellipses, etc.
And what does this have to do with associating all sets to geometric forms?

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

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?

 Quote by Hurkyl And what does this have to do with associating all sets to geometric forms?
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.

 Recognitions: Gold Member Science Advisor Staff Emeritus Although you're using the words, you don't seem to be doing mathematics, so I'll move this thread over here.

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