Simplicity, Symmetry, Consistency

November 17, 2004

Doron' date='

People use set theory because it has given years of fruitful mathematics. It also is just fine and dandy consistent. If you were serious about set theory, you'd learn something developed after 1930. There are plenty of great constructive set theories that fit your limitations on the concept of actual infinity, yet they do not deny the consistency of normal set theory. There are many constructive theorems about the limits of the applicabilty of ordinary set theory and its usefulness in generating concepts for discussion.

Try Bishop and Bridges book on constructive mathematics. They are able to succesfully introduce all the concepts in this thread constructively.[/quote']

1) I am not talking about any specific version of Set Theory.

2) I am talking about a fatal conceptual mistake that Cantor did when he used the idea of sets.

My argument about the Cantorian transfinite system (where aleph0 is not in N), is based on the most simple things that, in my opinion, we should care about when we define a consistent framework, and these things are:

Simplicity and Symmetry.

In this case, we do not think about quantity (fewest possible elements) but about simplicity.

Only the simplest thing can be considered as a building-block of some pure framework.

In the case of Emptiness, it is the lowest concept that cannot be manipulated by any framework that is based on information, and the language of Mathematics is first of all an information system, like any language, formal or informal.

When we have the lowest concept that cannot be manipulated by any framework, then if we want to save the simplicity of our framework, we cannot ignore anymore its internal symmetry.

So, if we use Emptiness, then in order to save the simplicity of our framework, we use symmetry, and define Fullness as the highest concept that cannot be manipulated by any framework.

By saving the simplicity and symmetry of our framework, we actually define its operational domain, where we can work and do interesting Math.

Cantor missed this important insight and created an asymmetrical framework that do not aware to the highest concept that cannot be manipulated by any framework, which is Fullness

And the result is the transfinite universe that ignore Actual infinity (which is both Fullness and Emptiness concepts).

When we look at this diagram http://www.geocities.com/complementarytheory/RiemannsLimits.pdf we can clearly and simply see that Actual infinity is an inevitable fundamental concept of any set theory, which uses Emptiness as one of its concepts.

Strictly speaking, any set theory that uses Emptiness, cannot ignore Fullness, in order to be consistent.

Since the Cantorian approach uses Emptiness but ignore Fullness, it cannot be a consistent framework, and this conclusion is stronger then any proof which is based on some axiomatic system.

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About Symmerty:

Some climes that Symmetry is: “The possibility of an object to be divided into two equal parts ..."

I think that this interpretation is deeper: (physics interpretation) "the property of being isotropic; having the same value when measured in different directions" (http://www.cogsci.princeton.edu/cgi-bin/webwn2.0?stage=1&word=symmetry) because the original meaning is:

"The same thing"

And "the same thing" is the identity of a thing to itself (which is the basic form of symmetry) or the "the same thing" can be extended to some common relation between different self-identities.

In all cases, the name of the game is Symmetry, and by this concept we can research how self-symmetries (or self identities) interact with each other, in order to expose their deeper common source of symmetry.

Any other interpretation of the Symmetry concept misses the full fruitfulness of this concept.

November 17, 2004

Doron' date='

People use set theory because it has given years of fruitful mathematics. It also is just fine and dandy consistent. If you were serious about set theory, you'd learn something developed after 1930. There are plenty of great constructive set theories that fit your limitations on the concept of actual infinity, yet they do not deny the consistency of normal set theory. There are many constructive theorems about the limits of the applicabilty of ordinary set theory and its usefulness in generating concepts for discussion.

Try Bishop and Bridges book on constructive mathematics. They are able to succesfully introduce all the concepts in this thread constructively.[/quote']

1) I am not talking about any specific version of Set Theory.

2) I am talking about a fatal conceptual mistake that Cantor did when he used the idea of sets.

My argument about the Cantorian transfinite system (where aleph0 is not in N), is based on the most simple things that, in my opinion, we should care about when we define a consistent framework, and these things are:

Simplicity and Symmetry.

In this case, we do not think about quantity (fewest possible elements) but about simplicity.

Only the simplest thing can be considered as a building-block of some pure framework.

In the case of Emptiness, it is the lowest concept that cannot be manipulated by any framework that is based on information, and the language of Mathematics is first of all an information system, like any language, formal or informal.

When we have the lowest concept that cannot be manipulated by any framework, then if we want to save the simplicity of our framework, we cannot ignore anymore its internal symmetry.

So, if we use Emptiness, then in order to save the simplicity of our framework, we use symmetry, and define Fullness as the highest concept that cannot be manipulated by any framework.

By saving the simplicity and symmetry of our framework, we actually define its operational domain, where we can work and do interesting Math.

Cantor missed this important insight and created an asymmetrical framework that do not aware to the highest concept that cannot be manipulated by any framework, which is Fullness

And the result is the transfinite universe that ignore Actual infinity (which is both Fullness and Emptiness concepts).

When we look at this diagram http://www.geocities.com/complementarytheory/RiemannsLimits.pdf we can clearly and simply see that Actual infinity is an inevitable fundamental concept of any set theory, which uses Emptiness as one of its concepts.

Strictly speaking, any set theory that uses Emptiness, cannot ignore Fullness, in order to be consistent.

Since the Cantorian approach uses Emptiness but ignore Fullness, it cannot be a consistent framework, and this conclusion is stronger then any proof which is based on some axiomatic system.

-------------------------------------------------------------------------------------------

About Symmerty:

Some climes that Symmetry is: “The possibility of an object to be divided into two equal parts ..."

I think that this interpretation is deeper: (physics interpretation) "the property of being isotropic; having the same value when measured in different directions" (http://www.cogsci.princeton.edu/cgi-bin/webwn2.0?stage=1&word=symmetry) because the original meaning is:

"The same thing"

And "the same thing" is the identity of a thing to itself (which is the basic form of symmetry) or the "the same thing" can be extended to some common relation between different self-identities.

In all cases, the name of the game is Symmetry, and by this concept we can research how self-symmetries (or self identities) interact with each other, in order to expose their deeper common source of symmetry.

Any other interpretation of the Symmetry concept misses the full fruitfulness of this concept.

matt grime · November 17, 2004

More opinions on mathematics, no facts, definitions or proofs.... wonder where we've seen this before....

Set theories do not ignore what you might mean by fullness, it is simply not a set in that theory (here I mean the mainstream ones), but a proper class. The collection of all sets (in some theory) is not a set in that theory. And? A vector space is a collection of vectors, but not itself a vector. Sorry you don't care to understand set theory, but that's not our fault, so please stop bombarding people with your misinterpretations of mathematics.

matt grime · November 17, 2004

More opinions on mathematics, no facts, definitions or proofs.... wonder where we've seen this before....

Set theories do not ignore what you might mean by fullness, it is simply not a set in that theory (here I mean the mainstream ones), but a proper class. The collection of all sets (in some theory) is not a set in that theory. And? A vector space is a collection of vectors, but not itself a vector. Sorry you don't care to understand set theory, but that's not our fault, so please stop bombarding people with your misinterpretations of mathematics.

premjan · November 17, 2004

I think symmetry is much less interesting than asymmetry. Symmetry is only a starting point to understanding real information, which consists mostly of meaningful asymmetries, once all possible symmetries have been factored out.

premjan · November 17, 2004

I think symmetry is much less interesting than asymmetry. Symmetry is only a starting point to understanding real information, which consists mostly of meaningful asymmetries, once all possible symmetries have been factored out.

Dave · November 17, 2004

I have to say, I'm finding this all rather tedious. Granted, I don't know a lot about set theory, but I do know how a piece of mathematics should read, and this just isn't maths tbh.

Please either post something using some strict mathematical notation or just don't post. I'm fed up of reading the same old thing time after time, quite frankly.

Dave · November 17, 2004

I have to say, I'm finding this all rather tedious. Granted, I don't know a lot about set theory, but I do know how a piece of mathematics should read, and this just isn't maths tbh.

Please either post something using some strict mathematical notation or just don't post. I'm fed up of reading the same old thing time after time, quite frankly.

MandrakeRoot · November 18, 2004

Yeah dave you said it alright.

THe writing just misses the mathematical rigor or even the scientific universality ....

Mandrake

MandrakeRoot · November 18, 2004

Yeah dave you said it alright.

THe writing just misses the mathematical rigor or even the scientific universality ....

Mandrake

premjan · November 18, 2004

however, I have to say that there are some good points in monadic (three-valued) logic, principally, the ability to have three truth values: true, false, and undefined (i.e. don't know). It could greatly reduce the number of propositions that have to be checked for verifying various logical expressions. Could be potentially be very useful in AI type situations, so perhaps mathematics is not really the right forum for this stuff.

premjan · November 18, 2004

however, I have to say that there are some good points in monadic (three-valued) logic, principally, the ability to have three truth values: true, false, and undefined (i.e. don't know). It could greatly reduce the number of propositions that have to be checked for verifying various logical expressions. Could be potentially be very useful in AI type situations, so perhaps mathematics is not really the right forum for this stuff.

matt grime · November 18, 2004

actually you increase the number of cases to check (obviously) if you allow three truth values.

AI uses lots of models of logic (neural nets, fuzzy etc) and there are whole journals dedicated to them.

Note to Doron. In category theory there is a category of categories, if one is careful.

matt grime · November 18, 2004

actually you increase the number of cases to check (obviously) if you allow three truth values.

AI uses lots of models of logic (neural nets, fuzzy etc) and there are whole journals dedicated to them.

Note to Doron. In category theory there is a category of categories, if one is careful.

November 18, 2004

The language of Mathematics can work only if there are elements that can be manipulated, and these elements can be used as inputs in our Mathematical tools.

There are two concepts that cannot be used as inputs, and they are:

Emptiness (too weak to be used as an input) and Fullness (too strong to be used as an input) and both of them are actual infinity.

What's left is point-like or segment-like building-blocks, that can be manipulated by the language of Mathematics.

If you take a collection of infinitely many point-like and/or segment-like building-blocks, and use a Universal-Quantification that is related to them, then you are no longer in any model that can be described by infinitely many elements, but only by Fullness, which is an unavailable information form that it is too strong to be manipulated by the language of Mathematics.

The model of fullness is described by an infinitely long (totally pointless) solid element.

There is no middle state between the infinitely_many_elements model and the infinitely_long_(totally_pointless)_solid_element model.

Since aleph0 is not in the domain of the infinitely_many_elements model, it cannot be but in the domain of the infinitely_long_(totally_pointless)_solid_element model.

In this case the transfinite system cannot be manipulated by the language of Mathematics.

Therefore the Cantorian transfinite universe does not hold.

By using the Reimaenn's circle model http://www.geocities.com/complement...mannsLimits.pdf I clearly and simply demonstrate these two models.

This model has nothing to do with geometry, it is a pure analytic model that clearly represents the number system itself, and also clearly shows its highest limitation.

Sets are used (in this case) to represent the same number system, so in both models we are talking about the same system.

But my representation is a better way to understand the number system, because through it we get the very impotent insight of the symmetry concept, which is totally ignored by the Standard set representation.

Standard Set system has a bottom information limitation, which is Emptiness, and I add to the same system its top information limitation, which is Fullness.

When I do that, the Standard Set system gets its necessary internal symmetry, that without it, it cannot be a consistent pure framework.

By using this symmetry, we clearly show that the Cantorian aleph0 does not hold.

Set theories do not ignore what you might mean by fullness...

Matt' date=' I'll say it again: Fullness is not a collection.

Fullness is represented by an infinitely_long_(totally_pointless)_solid_element model.

Therefore it is not the set of all sets (which is represented by an infinitely_many_elements model), please understand it, thank you.

I think symmetry is much less interesting than asymmetry.

I totally agree with you, but in order to get asymmetry we first have to break some symmetry.

THe writing just misses the mathematical rigor or even the scientific universality

The true values of my framework can be found in the truth tables in pages 4' date='5 of http://www.geocities.com/complementarytheory/CompLogic.pdf.

Also the symmetry of my framework can be found on page 6 of the above address.

I have to say that there are some good points in monadic (three-valued) logic,

Monadic Mathematics is not based on 3 valued logic, but on Included-middle logical reasoning, which is a serial/parralell_multi_valued logic.

November 18, 2004