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### #1 doronshadmi

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Posted 15 March 2008 - 12:09 AM

"Cantor introduced into mathematics the notion of a completed set, so that the integers, for example, could be considered together as a set in themselves, and so as a completed infinite magnitude. Only by conceiving of the integers as a whole entity, (as a Ding für sich) could Cantor define the first transfinite number, which he denoted by a lower case omega (ω), in contradistinction to the familiar sideways eight infinity symbol (∞), which had only meant unbounded." ( http://www.asa3.org/...3-93Hedman.html )

Cantor is wrong because no non-finite collection has the magnitude of the non-local ur-element real-line, and as a result any given non-finite collection is open and incomplete when compared with the non-local ur-element real-line's magnitude.

Furthermore, Cantor had the chance to formalize the actual infinity in terms of a non-local atom, but he mixed between religion
and logical reasoning and missed the notion of non-locality.

"Cantor is quoted as saying:

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."

( http://en.wikipedia....solute_Infinite )

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A hidden assumption :

Algebare is not a Symmerty\Asymmetry synthesis because all the terms in algebra are clearly defined ahead of time.

Time is not involved in Symmetry\Asymmetry synthesis as clearly shown by the universal principle that stands in the basis of the multitude, whether it is Logic or Algebra.

Here it is again:

Let a 2-valued framework be represented by A B.

A B system is:

A B
A A
A B
B A
B B

This system can be reduced (without a loss of generality) to
symmetric (AA or BB) \ asymmetric (AB or BA) synthesis, represented as:

AB
XX is AB symmetry --> A=B (AB is the same)
XY is AB asymmetry --> A≠B (AB is not the same)

A=B can be reduced (without a loss of generality) to a single value X.

A≠B cannot be reduced to a single value without a loss of detail X or Y.

Furthermore, in order to conclude that A≠B, they must share the same realm.

So 2-valued framework is at least X (symmetry or sameness) \ XY (asymmetry or difference) synthesis.

Let us use SA (Symmetry\Asymmetry synthesis) on 2-valued Logic:

T F NXOR
F F considered
F T not considered
T F not considered
T T considered

T F XOR
F F not considered
F T considered
T F considered
T T not considered

T F NXOR\XOR
F F considered
F T considered
T F considered
T T considered

T F generalization of NXOR\XOR
X X considered
X Y considered

As for algebra, it holds only if at least two things(Asymmetry) share the same realm (Symmetry).

...all the terms in algebra are clearly defined ahead of time

This conclusion, which is based on the concept of the Multitude, holds because of the Symmetry\Asymmetry shynthesis that enables this concept.

By NXOR\XOR logic we may fulfill Hilbert's organic paradigm of the mathematical language. Quoting Hilbert’s famous Paris 1900 lecture:

“…The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.”

A hidden assumption:

An interaction between different things (abstract or not) is possible, only if they share some common property known as their realm.

Without it each thing is totally disconnected from any other thing, and there is nothing beyond one.

XOR connective is the logical basis of disconnection where no more than a one thing exits.

If something is a one of many things, then its realm is not less than a relation between XOR (the logical basis of disconnection) and NXOR (the logical basis of connection).

A set is a NXOR\XOR realm product, because the quantifier "for all…" is used in addition to the quantifier "there exist …", for example:

The standard definition of a proper subset is:

A is a proper subset of B if for all x that are members of A, x are members of B but there exist a y that is a member of B but is not a member of A.

NXOR is used as a hidden assumption of the definition above. In order to see it, let us omit the quantifier "for all…" and we get:

A is a proper subset of B if x that is a member of A, x is a member of B but there exist a y that is a member of B but is not a member of A.

Let us examine this part:

but there exist a y that is a member of B but is not a member of A.

NXOR is used as a hidden assumption in two cases here:

Case 1: In order to distinguish x from y we need some relation between them that enables us to simultaneously compare one with the other, and this simultaneity is an NXOR connective.

Case 2: In order to distinguish A from B (and conclude that A is a proper subset B) we need some relation between them that enables us to simultaneously compare one with the other, and this simultaneity is an NXOR connective.

Simultaneity (in this case) means that A XOR B (or x XOR y) share the same realm (and it is timeless) and we get a NXOR\XOR realm.

A hidden assumption is devastating in the case of Logic and Mathematics.

NXOR:

NXOR is the logic that enables us to define the relations among abstract/non-abstract elements.

We will not find it in any book of modern Mathematics, because Modern Mathematics uses it as an hidden assumption.

NXOR is recognized as a property called memory, which enables us to connect things and research their relations.

Our natural ability to connect between objects (notated by ↔ and known as map, or function) is a hidden assumption of the current formal language.

It has to be understood that in order to define even a 1↔1 map, we need not less than a XOR product (notated as 1) and a NXOR product (notated as ↔).

Map is a connection (a NXOR product) that enables us to define the relations between more than a 1 XOR product, and we cannot go beyond 1 without ↔ between 1,1 .

The number 2 is actually the relations of our own memory as ↔ (as a function) between 1 object XOR 1 object.

Measurement:

" The earliest and most important examples are Jordan measure and Lebesgue measure,..."
(mathworld.wolfram/Measure)

Jordan measure:
"... The Jordan measure, when it exists, is the common value of the outer and inner (NXOR hidden assumption) Jordan measures of M"
(mathworld.wolfram/JordanMeasure)

So we need a common property in order to measure, and this common properly is based on NXOR connective that is related to XOR connective products, known as members.

So membership is not less than NXOR(the common) XOR(the distinct) relations, and (for example) the common value of set N is Size.

Lebesgue Measure:
" ... A unit line segment has Lebesgue measure 1; the Cantor set has Lebesgue measure 0. (mathworld.wolfram/LebesgueMeasure)

A segment is not less than A AND B (Lebesgue measure 1).

A set of disjoint elements (finite or non-finite) has a Lebesgue measure 0.

The Lebesgue measures 1 and 0 are equivalent to NXOR(non-local) and XOR(local) products of my system, but in the traditional system Lebesgue measures 1 is not an atom, but it is a XOR-only product (made of non-finite local elements).

It has to be understood that nothing can be measured beyond one without a relation between the local and the nonlocal (the concept of "many ..." does not exist without this relation.)

A proper subset: (a definition without a hidden assumption)

C is a proper subset of B only if both of them are based on property A and any C member is also a B member, but there is a B member that is not a C member.

For example: Size is a common property of both N and any proper subset of it.

Let E be any N member, which is divided by 2.

E is a proper subset of N only if the Size property is not ignored, so let us examine this mapping:
```E  = { 2,  4,  6,  8,  10,  12,  14,  16,  18,  ... }
↕   ↕	↕   ↕   ↕	↕	↕	↕	↕
N  = { 1,  2,  3,  4,   5,   6,   7,   8,   9,  ... }```
In the example above there is a 1-1 correspondence between E and N because we ignore Size as a common property of both N and E, and define the 1-1 correspondence between the notations that represent the size, by ignoring the size itself.

Here is the right mapping between E and N, where Size as a common property is not ignored:
```E = { 2,  4,  6,  8,  ... }
↕   ↕   ↕   ↕
N = { 1,  2,  3,  4,   5,   6,   7,   8,  ... }```

If (for example) notation 8 exists in E, then the size that it represents must exist in N, and only then E is a proper subset of N. By not ignoring Size as a common property of the natural numbers, we can clearly see that there is no 1-1 correspondence between E and N.

By the way, order is not important here, and the non-ordered mapping below is equivalent to the ordered map above:
```E  = { 8,  4,  2,  6,  ... }
↕   ↕   ↕   ↕
N  = { 7,  8,  3,  1,   6,   4,   2,   5,  ... }```

Conclusion:

Galilio and Dedeking made a simple mistake when they defined a 1-1 correspondence between notations by ignore the common property that they represent.

Cantor used this mistake in order to define the non-finite property of N by claiming that there is a 1-1 correspondence between N and a proper subset of it.

But as we show here, he was wrong in this case.

A non-finite set is simply a set that does not have a final member, and both E and N are non-finite sets as well, where Size_of_E/Size_of_N has a permanent ratio of 1/2.

Aleph0 from NXOR\XOR point of view:

Cantor's theorem about the Size of the non-finite is based on the notion that aleph0+1=aleph0, or in other words Cardinality (or Size) is not changed under addition when we deal with infinitely many objects.

By Cantor, a Size that is not 0 (he called it aleph0) is unchanged under addition.

I understand the Set concept also from a NXOR point of view.

From this additional point of view no XOR product (anything that it is "a one of many …", and the Set concept is based on it) can be an NXOR product, and as a result (which is based on logic, and not on intuition) the Size of any non-finite set is logically incomplete (it cannot be an NXOR product, no matter how infinitely many elements it has).

Since the Size of a non-finite set is incomplete we cannot use Dedekind's 1-1 method in order to define the exact sizes of two non-finite sets (each one of them is an incomplete mathematical object).

Instead, we define the proportion that exists between non-finite (and logically incomplete(XOR))products (which are called sets), and the permanent proportion of aleph0+1/aleph0 is a non-local number greater than 1 (Remark: NXOR is used as a hidden assumption of the standaed concept of Set, and as a result local-only members go beyond one).

If this proportion is important for us, we can use some notation in order to represent aleph0+1/aleph0 (for example: "Let @ be the representation of the permanent proportion aleph0+1/aleph0") but we must not mix between the notation "@" and the value that it represents, and Dedekind 1-1 method does not distinguish between a value and its representation, and defines the map between the representations instead of between the values themselves, as I show here.

A 1-1 correspondence: (a new point of view)

A 1-1 correspondence exists between two non-finite sets (as I show here) if each set is a collection of unique objects, where each object has nothing in common with the rest of the unique objects.

In that case the 1-1 mapping is between infinitely many separated objects.

In this case each mapping is disjoint from any other mapping, and we get the ratio of 1/1 which is equivalent to a 1-1 correspondence.

But then infinitely many disjoint 1-1 mappings cannot be considered as a mapping between a set and its proper subset (because each mapping is a separated case) and all we have is infinitely many separated cases, with a 1/1 ratio.

Edited by doronshadmi, 15 March 2008 - 12:10 AM.

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### #2 Duh Puck

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Posted 15 March 2008 - 06:11 AM

This is a very interesting topic, but my first reaction was that it didn't have much practicality. However, the more I think about it, the more I can see the connection to philosophical and scientific application. I look forward to giving greater consideration to your suggestions when I have more time.

However, I would first like to thank you for the link to the ASA site. Just a quick sampling revealed a wealth of interesting articles.
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### #3 bonanova

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Posted 15 March 2008 - 08:47 AM

An interaction between different things (abstract or not) is possible, only if they share some common property known as their realm.
Without it each thing is totally disconnected from any other thing, and there is nothing beyond one.

True, true. I knew a woman like that, once.
Once was enough.
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Vidi vici veni.

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