Possible common base ground for both Set and Category Theories (1)

[doronshadmi]

I wish to share with you some of my late developments, which study the possible common base ground for both Set and Category Theories.

Here is a part of the paper called "Organic Mathematics (A Non-Formal Introduction)" http://www.geocities.com/complementarytheory/OMPT.pdf :

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2. Cardinality Sets and Categories

Let X be a placeholder for any thinkable thing. X can be measured by using Set as a measurement tool, where Cardinal is the measurement unit. For example, the ZF axiom of the Empty Set states that: "There exists set A such that any set (including A) is not a member of A". By OM this axiom is understood as follows: "There exists set A" means that if set A is measured as a member of some set, for example B={A}, then the cardinal of B is at least 1. If we generalize it to "There exists set", then the magnitude of the existence of a Set is at least 1. Following the same reasoning, the magnitude of the existence of Emptiness is 0, where the magnitude of the existence of its opposite, called Fullness, is ∞. In each one of these examples, Set is used to measure the magnitude of the existence of X.

In the case of the Empty set, X is the absence of members. In the case of the Full set, X is stronger than any member. "X is stronger than any member" is what OM calls Relation, known as "Membership" by Set Theory or "Morphisms" by Category Theory.

In both theories Collection is the result of Relation Element Interaction (REI), where the cardinality (the magnitude of the existence) of this result is > 0 and < ∞.

The magnitude of the existence of a set is not determined by its members if these "members" are Emptiness or Fullness. Emptiness or Fullness are not researchable directly, because Emptiness' existence on its own is too weak, and Fullness' existence on its own is too strong. For example, The Empty Set is not itself Emptiness but it is an existing thing that is used to define the Cardinal of Emptiness, where Emptiness' "existence" itself is weaker than any existing thing. Also the Full Set (the opposite of the Empty set) is not itself Fullness but it is an existing thing that is used to define the Cardinal of Fullness, where Fullness' "existence" itself is stronger than any existing thing.

In that case the concept of Set has a magnitude of existence that is stronger than 0 (the magnitude of the "existence" of Emptiness that can be defined only indirectly by using an existing and researchable thing like Set) and weaker than ∞ (the magnitude of the "existence" of Fullness that can be defined only indirectly by using an existing and researchable thing like Set). By getting the notion of the extreme non-researchable states (Emptiness or Fullness on their own) one defines the general concept of Collection, where its magnitude of existence is stronger than Emptiness on its own and weaker than Fullness on its own.

If we generalize Sets or Categories by these notions, then Memberships (Set) or Morphisms (Category) magnitude of existence are weaker than Fullness on its own and stronger than Members (Set) or Objects, where Members (Set) or Objects (Category) magnitude of existence are stronger than Emptiness on its own and weaker than Memberships (Set) or Morphisms (Category). If Memberships or Morphisms are Relation and Members or Objects are Element, then the magnitude of existence of a non-empty collection is determined by the amount of its Elements, gathered by Relation.

In order to distinguish between the researchable and the non-researchable, let us symbolize it as follows:

Emptiness on its own is represented by the background of this page.

Fullness on its own is represented by the opposite background of this page.

Relation is represented as _

Element is represented as •

Interaction (Bridging) between Relation and Element is represented as |

( For further reading, please look at http://www.geocities.com/complementarytheory/OMPT.pdf )

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[doronshadmi]

OM 's development is possible because we determine the limits of the researchable by using the weak limit (Emptiness) and the strong limit (Fullness). Cantor distinguished three levels of existences:

1) In the mind of God (the Intellectus Divinum)

2) In the mind of man (in abstracto)

3) In the physical universe (in concreto)

By using Fullness as "that has no successor" we show that Cantor's in abstracto Transfinite system is not an actual infinity. We also show how Distinction is a first-order property of any collection. These developments are based on a cognitive approach of the mathematical science. In "On the Reality of the Continuum" [1] (page 124) we find this sentence:

"From the realist standpoint, numbers and other real things do not need admitting or legitimating by humans to come into existence."

From the idealist standpoint, numbers and other real things do need admitting or legitimating by humans to come into existence. In both cases the term "real thing" has to be understood. According to the realist if "real things" are "real" iff they are totally independent of each other, then no collection is a "real thing" (total independency does not able things to be gathered).

According to the idealist if "real things" are "real" iff they are totally dependent of each other, then no collection is a "real thing" (total dependency does nor able things to be identified). No collection exists in terms of total dependency (total connectivity) or total independency (total isolation). Since totalities are not researchable on their own, then any research cannot avoid the existence of collections, where collections are the only researchable "real things". Actually we find that a researchable realm is both ideal (has relations) and real (has elements).

We have to notice that there is no symmetry in using concepts like "Realist standpoint" in order to understand "real things" because if the requested result is "real things" then we actually give a privilege to the Realist standpoint over the Idealist standpoint about the requested "real thing". This asymmetry can be avoided by changing the requested results to "researchable things" instead of "real things". In that case the concept of Collection is researchable exactly because it is not totally real and not totally ideal.

Here is the last part of the quote from [1]:

"Furthermore, real objects are always legitimate objects of study in the sciences, even if they are not fully understood or known."

We agree with this quote because "real objects" are valuable for science iff they are researchable, or in other words, they are both real and ideal.

For further clarification of my work with Moshe Klein on this subject, please see the attached links:

PowerPoint first presentation:

http://www.geocities.com/complementarytheory/ONN.pps

PowerPoint advanced presentation (slides clarifications):

http://www.geocities.com/complementarytheory/OM-PPT2.pdf

PowerPoint advanced presentation:

http://www.geocities.com/complementarytheory/OM-PPT2.pps

I will appreciate very much you reply.

[1] Anne Newstead & James Franklin, On the Reality of the Continuum Philosophy 83
( 2008 ), 117-27 http://web.maths.unsw.edu.au/~jim/newsteadcontinuum.pdf .

[ddt]

So instead of answering the questions in the JREF thread you started, you run away to another forum?

[Graybeard]

Can I believe my eyes. Are you two carrying out a turf war on our territory ?

Graybeard (mod)

[ddt]

Can I believe my eyes. Are you two carrying out a turf war on our territory ?

Graybeard (mod)

Turfwar? Not really. Just want to warn you that doronshadmi has been peddling his crackpot "mathematical" ramblings now on over 50 forums. Doron's understanding of math is invariably profoundly wrong. Just google for his name and check out some of his earlier threads.

[doronshadmi]

By OM, Cardinal is a measurment unit of the exictence of a thing where Set is an existing measurment tool that its magnitude of existence is > 0 and < ∞.

Michael Potter in his excellent book Set Theory and its Philosophy ( http://books.google.com/books?id=FxRoPuPbGgUC&printsec=frontcover&source=g bs_similarbooks_r&cad=4_2#PPA21,M1 ) clearly explains in chapter 2 the difference between collection and fusion.

If Cardinality is used to measure the existence of what he calls aggregation, then:

1) If aggregation means fusion and the cardinal of an aggregation = 0, then no aggregation exists.

2) If aggregation means collection and the cardinal of an aggregation = 0, then there exists an aggregation with no members.

Let us think about the opposite cardinal 0, called cardinal ∞ .

Again, if Cardinality is used to measure the existence of what he calls aggregation, then:

3) If aggregation means fusion and the cardinal of an aggregation = ∞, then no aggregation exists because the existence of a fusion that has cardinal ∞ is stronger than the existence of aggregation, exactly as the existence of a fusion that has cardinal 0 is weaker than the existence of aggregation.

4) If aggregation means collection and the cardinal of an aggregation = ∞, then no members belong to the collection that has cardinal ∞ . Fullness is stronger than the existence of members, exactly as Emptiness is weaker than the existence of members.

Fusion is not researchable in the case of cardinal 0 or cardinal ∞, because Emptiness' existence is too weak and Fullness' existence is too strong.

On the contrary collection is researchable in the case of cardinal 0 or cardinal ∞, because the collection's existence has a cardinal that is
> 0 and < ∞.

In that case no collection of non-finite members is complete (its cardinal < ∞), and as a result its exact cardinality is not well-founded. In that case the universal quantifier "for all" does not hold in the case of non-finite collections.

A finite collection has an accurate cardinal because its magnitude of existence is well-founded. In order to understand it we have to use Distinction as a first-order property.

This is exactly what I did in "Organic Mathematics (A Non-Formal Introduction)" http://www.geocities.com/complementarytheory/OMPT.pdf

[doronshadmi]

Some claims that we cannot use numbers in defining sets since sets are used in defining numbers, or in other words, we are using here a circular reasoning.

Let's carefully investigate this claim.

This claim holds if one does not distinguish between collections and fusions.

In this case one cannot get the difference between:

1) |{X}| = the measurement unit of X existence, where X is a fusion (in the case of Emptiness or Fullness) or X is a collection (in the case of a set).

2) {X} as an existing measurement tool of X, where {X} is a collection, and X is a fusion or a collection.

So |{X}| measures the existence of the members of {X} and does not measure the existence of {X}.

In other words, there is no circularity here because the cardinal measures the existence of the member of {X}, and it does not measure the existence of {X}.

If one forcing {X} = 1, then one uses a circular reasoning..

I am talking about |{X}| = 1 (where X is a fusion -in the case of Emptiness or Fullness- or X is a collection -in the case of a set-), which is not circular, because |{X}| is not {X}.

[doronshadmi]

Can I believe my eyes. Are you two carrying out a turf war on our territory ?

Graybeard (mod)

Hi Graybeard,

Thank you for the "warm" wellcome.

Please explain who is exactly these "our"?

[ddt]

Some claims that we cannot use numbers in defining sets since sets are used in defining numbers, or in other words, we are using here a circular reasoning.

Strange remark: no-one here claimed that. But on JREF, jsfisher did that. Can't keep track of what you post where?

You may also note that doron posts exactly the same posts here and on JREF (thread in above link) and on talkrational. Tell us, doron, why do you seek out new fora every time? And why do you typically spam your nonsense on two, three fora at the same time?

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Please explain who is exactly these "our"?

Guess who that would be - at the time of Graybeard's post, only two members had posted in this thread. I won't conceal the fact I just followed you hear to be able to warn the crowd over here for the vacuousness of your posts.

[doronshadmi]

Strange remark: no-one here claimed that. But on JREF, jsfisher did that. Can't keep track of what you post where?

You may also note that doron posts exactly the same posts here and on JREF (thread in above link) and on talkrational. Tell us, doron, why do you seek out new fora every time? And why do you typically spam your nonsense on two, three fora at the same time?


Guess who that would be - at the time of Graybeard's post, only two members had posted in this thread. I won't conceal the fact I just followed you hear to be able to warn the crowd over here for the vacuousness of your posts.

ddt tries to hijack the dialog of this thread to another forum.

I suggest to avoid from him in doing it.

Generally ddt is obsessed about my actions in the internet and makes an history list of these actions.