# A proof that |R| < C

Because by Conventional-Mathematics, every R member is a point in the Real-line, then:

{.} = any non-empty R member

|{}| = The cardinality of the empty set = 0

|{.}| = The smallest catdinality of any non-empty set = 1

0 or 1 are members of W.

Between 0 to 1 there is {} or in other words, the transition from
0 to 1 and vice versa, is a phase transition or a quantum leap.

The transition's type from {} to {.} = the transition's type from 0 to 1 but,
because it is a phase transition, and we cannot use {} or {.}
as an element between {} to {.}, we have no choice but to
define a new set's content.

{} = Emptiness
{.}= Localized element = Point
{_}= Non-localized element = Line

Therefore between {} to {.} there is {_}.

Now we can conclude that between any two non-empty R members there
exist a non-localized element.

Therefore |R| does not have the power of the Continuum.

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Integral
Staff Emeritus
Gold Member
Because by Conventional-Mathematics, every R member is a point in the Real-line, then:

{.} = any non-empty R member

|{}| = 0

|{.}| = 1

There lies your trouble, as far as the real numbers go the only difference between 1 and 0 is location. You seem to be implying that 0 is somehow a "special" spot in the real line. It is only special in that it is defined to be the additive idenity element. So if (in your notation
|{.}| = 1

is a point on the real line then we must also have

|{.}| = 0

Since 0 is simply a point on the real line like every other point on the real line.

If you can get a hold of that fact perhaps you can quit wasting your time with these nonsense "proofs"

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HallsofIvy
Homework Helper
As long as you insist upon changing the definitions of the words, you can "prove" anything you like but it will still be meaningless.

I could easily prove that C< |R| by C to be the number of tables in my house and |R| to the number of chairs! Based on MY definitions, C< |R| but it is as meaningless as your post.

Hi Integral,

How can the cardinality of a non-empty R member can be less than 1 ?

|{}| = the cardinality of the empty set = 0

The cardinality of any non-empty set ( |{.}| ) never can be less then 1 !

So, |{.}| = 0 simply does not hold.

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

Hi HallsofIvy,

A table or a chair or any other element that is not nothingness, cannot have a cardinality, which is less than 1 .

So, your argument is based only on the non-empty word, but my proof is based on both empty and the non-empty words.

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Integral
Staff Emeritus
Gold Member
Not only do you make up your own definitions, you change them at the whim of the winds.

It would be time well spent for you to find a good book on Real Analysis. You need to make an effort to learn what how to communicate your ideas. The best way to do this is within the frame work which already exists. It is a waste of time to reinvent the wheel. Espesially since you seem to want a square one.

Edit:
Just where is this emptiness,or gap< you continue to speak of? Is this gap unique to zero? Why? Why does the gap not exist between any 2 integers? What happens if I compute {gap} + 1? or 10? Why are you so intent on finding this gap at zero? Why not at 100?

I, and other, mathematicains will point out that a gap does not exist, since we can place a number in the middle of any gap you name. So where is this gap? How can a gap, which is filled with numbers be a gap?

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Sorry Integral,

But you did not answer my argument.

All what I did is to tune my proof, after I realized that you don't understand it.

Yours,

Doron

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Integral
Staff Emeritus
Gold Member
You have not answered any of my questions either.

Integral,

It is my mistake, I forgot to write that 0 or 1 are the members of W.

So, now you will understand why I put {} between 0 to 1, in the first stage of my proof.

A proof that |R| < C
--------------------

Because by Conventional-Mathematics, every R member is a point in the Real-line, then:

{.} = any non-empty R member

|{}| = The cardinality of the empty set = 0

|{.}| = The smallest catdinality of any non-empty set = 1

0 or 1 are members of W.

Between 0 to 1 there is {} or in other words, the transition from
0 to 1 and vice versa, is a phase transition or a quantum leap.

The transition's type from {} to {.} = the transition's type from 0 to 1 but,
because it is a phase transition, and we cannot use {} or {.}
as an element between {} to {.}, we have no choice but to
define a new set's content.

{} = Emptiness
{.}= Localized element = Point
{_}= Non-localized element = Line

Therefore between {} to {.} there is {_}.

Now we can conclude that between any two non-empty R members there
exist a non-localized element.

Therefore |R| does not have the power of the Continuum.

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Integral
Staff Emeritus
Gold Member
Integral,

It is my mistake, I forgot to write that 0 or 1 are the members of W.

What is W?
So, now you will understand why I put {} between 0 to 1, in the first stage of my proof.
What proof? No, I do not understand what {} is.

A proof that |R| < C
--------------------

Because by Conventional-Mathematics, every R member is a point in the Real-line, then:

{.} = any non-empty R member
What is a non-empty R member?

|{}| = The cardinality of the empty set = 0

|{.}| = The smallest catdinality of any non-empty set = 1

0 or 1 are members of W.
What is W?
Between 0 to 1 there is {} or in other words, the transition from
0 to 1 and vice versa, is a phase transition or a quantum leap.
This is what you are trying to prove. Where is the proof? All you do is state it. Where is your definition of phase transition or Quantum leap?

The transition's type from {} to {.} = the transition's type from 0 to 1 but,
because it is a phase transition, and we cannot use {} or {.}
as an element between {} to {.}, we have no choice but to
define a new set's content.

{} = Emptiness
{.}= Localized element = Point
{_}= Non-localized element = Line

Therefore between {} to {.} there is {_}.
I see, you define it to be so. You have not defined "line element" what is that?
Now we can conclude that between any two non-empty R members there
exist a non-localized element.

Therefore |R| does not have the power of the Continuum.

Quite a leap, How did you make it? You have not given me any proof, only claims. You apparently have defined it to be true.

Where is this gap? Why is it only near zero?

What is W?
W is the set of the Whole numbers.

N = The set of all positive integers, whitout 0.
W = The set of all positive integers, include 0.
Z = The set of all integers.
Q = The set of all rational numbers.
R = The set of all irational numbers.
C = The set of all complex numbers.

Any memeber of the sets above, is a point = {.} in the real-line.
What is {} ?

{} = The Empty set (Set's content does not exist)
{.} = Some set's member (content exists)

|{}| = The cardinality of the Empty set = 0
|{.}| = The smallest catdinality of any non-empty set = 1

Between any two W members there is nothing(={}),and if there is, thay are not N,W,or Z members, but R or C members.

Therefore, there is a phase transition between |{}|(=0) to |{.}|(=1) and vice versa.

Between {}(= no content) to {.}{= a content) there is also a phase transition between {} to {.}, and vice versa.

We cannot use {} or {.} as an element between {} to {.}, so we have no choice but to define a new set's content.

{} = Emptiness
{.}= Localized element = Point
{_}= Non-localized element = Line

Therefore between {} to {.} there is {_}.

Now we can conclude that between any two non-empty R members there
exist a non-localized element.

Therefore |R| does not have the power of the Continuum.

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Integral
Staff Emeritus
Gold Member
W is the set of the Whole numbers

N = The set of all positive integers, whitout 0.
W = The set of all positive integers, include 0.
Z = The set of all integers.
Q = The set of all rational numbers.
R = The set of all irational numbers.
C = The set of all complex numbers.

Any memeber of the sets above, is a point = {.} in the real-line.

{} = The Empty set (Set's content does not exist)
{.} = Some set's member (content exists)

|{}| = The cardinality of the Empty set = 0
|{.}| = The smallest catdinality of any non-empty set = 1

Between any two W members there is nothing(={}),and if there is, thay are not N,W,or Z members, but R or C members.
This needs to be proven.
Therefore, there is a phase transition between |{}|(=0) to |{.}|(=1) and vice versa.
This needs to be proven.
Between {}(= no content) to {.}{= a content) there is also a phase transition between {} to {.}, and vice versa.
This needs to be proven.
We cannot use {} or {.} as an element between {} to {.},and we have no choice but to define a new set's content.
Prove it.
{} = Emptiness
{.}= Localized element = Point
{_}= Non-localized element = Line

Therefore between {} to {.} there is {_}.
You still have not provided a meaningful definition of "Non-localized element = Line" What is a line?
Now we can conclude that between any two non-empty R members there
exist a non-localized element.

Therefore |R| does not have the power of the Continuum.

You have proven nothing, you can conclude nothing.

Hi Integral,

Please give me more detailed information on one of the parts that you say
that it is has to be proven.

Thank you.

Integral
Staff Emeritus
Gold Member
What do you mean, more detail? What you say cannot be concluded from the infromation you have provided. Pleade proof these statements. A proof is not a series of unspported statements. It is a logical development where a fact follows from fact. What you present is a series of unsupported statement which do not follow from what goes before. Your terms are undefined, your conclusions are predetermined and by definition, not proof. You define a gap and call that proof. Not the case.

When will you answer a single question I ask. What do you mean by a line?

OK Integral,

I think I have got the idea, which is, when one does not understand something, then he wants it to be proven.

A line is a non-localized element, the opposite of a point, which is a localized element.

For example: an X-axis with no Y data is a line, or a non-localized element.

Any localized element (some point) needs at least x,y data.

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Integral
Staff Emeritus
Gold Member
OK Integral,

I think I have got the idea, which is, when one does not understand something, then he wants it to be proven.

Is there something wrong with that? If the connction was clear, and what you said was obvious then it would not need proof.
That is not the case for your statements.

A line is a non-localized element, the opposite of a point, which is a localized element.

For example: an X-axis without with no Y data is a line or a non-localized element.
I will not permit you define you line in terms of real numbers, you have already said that it is an absence of real numbers so how can it be complsed of point on the line. If it is not points WHAT IS IT!
Any localized element (some point) needs at least x,y data.

I still do not know what you are talking about. PLEASE SHOW ME WHERE A GAP IS

I will not permit you define you line in terms of real numbers, you have already said that it is an absence of real numbers so how can it be complsed of point on the line. If it is not points WHAT IS IT!

This is the Continuum itself, and it does not need any POINTS to exist.

We can use it as a connector (not a conteiner) between any two points.

In Conventional Math 0^0 is not well defined, because each number is a point (a localized element).

Let us say that power 0 is the simplest level of existence of some set's content.

Because there are no points in the Continuum, its base value = 0, but because it is exist (unlike the emptiness), its cardinality = 0^0 = 1.

|{}| = 0
|{_}| = 0^0 = 1
|{.}| = 1^0 = 1

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Integral
Staff Emeritus
Gold Member
Once again, you are changing the topic.

What is your continum, I do not understand this out of the context of the real numbers.

Please specify, WHERE IS THE 'liNE' Or GAP, are they the same thing or different.

You have steadfastly refused to answer this question. If you cannot show me where it is, IT DOES NOT EXIST.

Integral,

In Quantum mechanics there exist two opposite forms: a Particle and a Wave.

A Particle is a localized element, and a Wave is a non-localized element.

As you know, there is an information's XOR ratio between a Particle and a Wave.

I discoverd that the same information's XOR ratio exists between a Point and a Line.

So, you cannot noticed the Continuum, if you try to understand it in terms of its
opposite, which is a Point.

XOR ratio between LINES to POINTS
---------------------------------
0(LINE) 0(POINT) -> 0-(No information) -> no conclusion.
0(LINE) 1(POINT) -> 1-(Clear Particle-like information) -> conclusions on points.
1(LINE) 0(POINT) -> 1-(Clear Wave-like information) -> conclusions on lines.
1(LINE) 1(POINT) -> 0-(No clear information) -> no conclusion.

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Integral
Staff Emeritus
Gold Member
The entire point of a proof is to prove something. You only make claims. Please provide some proof and some definitions which will give your claims some meaning.

When did we jump from real analysis to Quantum Mechanics. Please stick to a single topic. Seems to me you are dancing around avioding the questions I ask. You have not been able to prove the existance of this thing you call a line element, whatever that is, you have not been able to prove the existance of a gap or quantum leap in the real line, you have proven nothing only made wild unsubstainicated claims. The entire point of a proof is to prove something, you do not have a clue what that means.

I am being to think you should not be allowed to post this nonsense in the math forum. Were it within my powers, I would move this thread to the Theory Development. Since you refuse to adhear to the known priciples of mathematics, this is not math. It is a thing of your own construction. You have not been able to demonstrate that it is either consistent or meaningful so what is the purpose?

Until you have done me the curtsy of making some effort to learn the universal language, I can see no real benifit in continuing this conversation.

Dear Integral,

Let us say that I have a new fundamental Idea, which is out of the conventional formal editing tools, and let us say that you are an editor.

First you have to understand the new fundamental idea.

After you understand it, you can accept or reject it.

If you accept it, you can try to edit it but, because it is a new fundamental idea (and you understand it), you realize that you have to develop a new editing tools that are out of the conventional formal editing tools.

You can't understand the new fundamental idea, if you try to understand it through the conventional formal system.

I gave you all what you need to do the step and see something new, but you did not want to do this step, which is OK, but than you can say nothing on my new idea, because you don’t understand it.

Have a good time.

Yours,

Doron

The convitional editing tools

Integral
Staff Emeritus
Gold Member
Thing is this is NOT a new fundamental Idea this is simply incorrect. There is NO NEED of your concocted line element.The real numbers are dense. The system is complete and it works.

BEFORE YOU CAN THINK OUT OF THE BOX YOU MUST KNOW WHERE THE BOX IS.

There are no gaps, no quantum leaps between numbers and no need of a simplistic line element.

You have no concept of what the box is or where the sides of it are. You are wasting your time with this nonsense.

IF you wish to develop an new number system do not base it on the reals, you are doing something different, NON MATHEMATICAL.

My Dear Integral,

Here is a more simple version of my proof:

Instead of using numbers, I do the simplest thing that I can do with a content of a set, which is, to explore its structure.

The general structure of any number is a point, which is a localized element.

So in this stage we have only 2 possible set's contents:

{} = Nothing

{.}= Point (something)

Now I ask: "what structure can exists between Point and Nothing ?"

The transition between Nothing to something cannot be but a phase transition.

In any phase transition you cannot put in between any of the already used elements.

Therefore, we cannot use Nothing nor Point as the element between {} to {.} .

In this stage we know that it cannot be Nothing and it cannot be a Point.

It means that we have to find some set's content, which is not Nothing and not a Point.

A point is a localized element and our new set's content must exist, because it can't be Nothing.

The only exists content, which is not a localized element, can't be but a non-localized element, as the new set's content.

And a non-localized element can't be but a Line.

Therefore we have found a new set's structure, which is {_} = Line.

A line is a non-localized element, therefore its structure is the opposite structure of a Point, which is a localized element.

It means that there is a XOR ratio between Lines to Points.

Therefore, no Line is built from Points.

Any number has the structure of a Point, therefore no line is built from numbers.

R is a set of numbers, and any number is a localized element.

Therefore we can conclude that there are exactly 0 R members in the path of some line.

Therefore |R| does not have the power of the continuum.

QED.

If now you understand the new idea, I'll be glad if you address it in a formal way.

Have a good day.

Yours,

Doron

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Hurkyl
Staff Emeritus
Gold Member
Instead of using numbers, I do the simplest thing that I can do with a content of a set, which is, to explore its structure.

Please explain what the "content of a set" has to do with "numbers"

The general structure of any number is a point, which is a localized element.

Where does this statement come from? What meaning could it possibly have? The words "point" and "localized" don't even exist in the language of the real numbers and set theory. Give these words a precise meaning.

So in this stage we have only 2 possible set's contents:

{} = Nothing

{.}= Point (something)

Highly imprecise; you seem to be saying sets either contain nothing or only a single object... which is certainly not true because sets can contain more than one element. I'm just guessing at what you might mean by "{.}"

Also, there's that "point" word again. Is this a definition of a point? What does this definition have to do with numbers?

Now I ask: "what structure can exists between Point and Nothing ?"

There's point again... still don't know what it means... but now you've added two new terms, "structure" and "between". I have no clue what a structure is.

"Between" is typically reserved for cases when there is some sort of ordering on the objects in question, but you have suggested no ordering, so I have no clue what "between" means.

The transition between Nothing to something cannot be but a phase transition.

Gak, two more terms! What is a transition? What is a phase transition?

In any phase transition you cannot put ____ in between any of the already used elements.

(underscores mine)

This statement isn't even grammatically correct; it's missing a direct object. Is this supposed to be a definition? If not, why should this be so?

Therefore, we cannot use Nothing nor Point as the element between {} to {.} .

Again, we don't know what "point" or "{.}" mean. You're making two assertions here, and I have no clue why either of them should be true.

Assertion 1: There exists a unique element (element of what?) between "{}" to "{.}". (did you mean to say "and" and not "to"?)

Assertion 2: neither "{}" nor "{.}" is that unique element

In this stage we know that it cannot be Nothing and it cannot be a Point.

Given your previous two assertions, correct. However, I don't know why your previous two assertions should be correct.

It means that we have to find some set's content, which is not Nothing and not a Point.

Why? What do you mean by a "set's content"? Why should the unique element between "{}" and "{.}" be a set's content? (Again, I still don't know what "between" means)

A point is a localized element and our new set's content must exist, because it can't be Nothing.

I recall you asserting a point is a localized element (though I don't know what that means or why it should be true). You are now making the assertion:

"A set's content cannot be {}"

The only exists content, which is not a localized element, can't be but a non-localized element, as the new set's content.

Due to the grammatical mistakes, I cannot decipher the overal structure of this sentence. I gleam an assertion that there is something called "content" (is this a "set's content"?) which cannot be a "localized element", and it cannot be a "non-localized element". Why should any of that be true? And of what new set's content are you speaking?

And what is a "non-localized element"?

And a non-localized element can't be but a Line.

Why is this true? What is a line?

Therefore we have found a new set's structure, which is {_} = Line.

So "{_}" is a structure named "Line". I presume that "{}" and "{.}" are also supposed to be structures named "Nothing" and "Point" respectively (And "Point" = "Something")

Are these the only "structures"? Does "structure" have any meaning beyond simply having the name "Line", "Point", or "Nothing"?

Are "." and "_" actual named objects, and is "{.}" supposed to be the set containing ".", and is "{_}" supposed to be the set containing "_"?

A line is a non-localized element, therefore its structure is the opposite structure of a Point, which is a localized element.

Why is a "line" a "non-localized element"? (I still want to know what both of those are supposed to mean)

Why does that imply its structure is the opposite structure of a point?

I thought "Line" was a "structure"... are you now saying "Line" has a "structure"?

What is an "opposite structure"? And how do you determinte the "opposite structure" of a "Point" is the "structure" of a "line"?

Before you were using "Point" as if it was a single named object, but now you use the phrase "a Point"... which is it?

It means that there is a XOR ratio between Lines to Points.

Are you saying something is a "Line" if and only if it is not a "Point"?

In other words, everything can be called either a "line" or a "point"?

Therefore, no Line is built from Points.

How does this follow? And what does "built" mean?

Any number has the structure of a Point, therefore no line is built from numbers.

What is a "number" and why would it have the "structure" of "a Point"?

You state that numbers have the structure of a Point.
You state that no Line is built from Points.

From these statements alone it does not follow that "no Line is built from numbers".

R is a set of numbers, and any number is a localized element.

Why can you say that the elements of R are "localized elements"?

Therefore we can conclude that there are exactly 0 R members in the path of some line.

How? And what is a "path" of a "line"?

Therefore |R| does not have the power of the continuum.

How does this follow?

Integral
Staff Emeritus
Gold Member
Therefore |R| does not have the power of the continuum.
Comtemplate this result, it is very interesting.
What does continuum mean? To most mathematicians it is the word used to descripe the cardinality of sets which have the same cardinality as the Reals. To be specific, the word continuum is defined to mean the cardinality of the Reals.

So you have proven that the Reals do not have the cardinality of the Reals!!!!

A wonderful result. Which, of course, means all that you have said is nonsense.

QED

Dear Intergral,

I really want to thank you.

You make a wonderful work with me, by giving me the opportunity to learn how a mathematician see my ideas (which in this stage looks to you like nonsense).

Sorry about my English mistakes, but Hebrew is my language, so I do the best I can, and from time to time I make a wrong translation from Hebrew to English.

Please be aware to the fact that I started to write English posts only about 1/2 year ago.

About your last post, and please tell me if I am wrong, mathematicians used the word Continuum as the power of Reals because through their conventional point of view (real analysis) they have found that no gap exist between any two R members.

The real analysis is based on two structural concepts of set's contents:

{} = Nothing

{.} = Point = well defined "sharp" element (a localized element) = something

Through this point of view, any non-empty set must have a structural form that is constructed from points.

Therefore, the original lexicographical meaning of the word Continuum (that can never be described as constructed from "Infinitely many ..." ) is changed to something that is constructed from "infinitely many points with no gaps between them".

My point of view is a return to the original lexicographical meaning of the Continuum, by adding it as a legitimate structural set's content, which is not{} and not{.} .

Believe me, it is a hard work for me, but I think that it must be done.

I rearranged my overview on the new theory of numbers, and now there are no "proofs" there, but clear (I hpoe) arranagment of my ideas on this subject.

You can find the overview here: http://www.geocities.com/complementarytheory/CATpage.html