Epsilon = Invariant Proportion

[lama]

Epsilon = Invariant Proportion

 

 

About 3.14... = circumference/diameter:

 

Let us say that Epsilon is equivalent to the invariant proportion that can be found in the triangles below.

 

(VERY IMPORTANT:

When Epsilon = Invariant Proportion, then there is no connection to words like 'smaller' or 'bigger' or 'size' or 'magnitude' or 'Quantity', and the reason is clearly explained)

 

,
|\
| \
|  \
|   \
|    |
|    |\
|    | \
|    |  \
|    |   \
|    |    |
|    |    |\
|    |    | \
|    |    |  \
|    |    |   |
|    |    |   |\
|    |    |   | \
|    |    |   |  |
|    |    |   |  |\
|____|____|___|__|_\

Each arbitrary right triangle's area is smaller than any arbitrary left triangle's area, but the internal proportion of each triangle remains unchanged, so it does not depend on size or magnitude (please think about circumference/diameter ratio, which does not depend on a circle's size).

 

If we have finitely many triangles then this proportion can be found finitely many times.

 

But in the case of infinitely many triangles, this proportion can be found infinitely many times.

 

Since Epsilon is equivalent to this proportion, it cannot be found if and only if this proportion cannot be found.

 

It is clear that if the proportion can be found infinitely many times, than it cannot be eliminated, and if it is eliminated, it means that it is found only finitely many times.

 

In other words, any collection of infinitely many elements can be found if and only if some epsilon that belongs to it also can be found, and if this Epsilon cannot be found, then there are only two options, which are:

 

a) The collection does not exist.

 

b) The collection is a finite collection.

 

 

Conclusion:

 

There is an inseparable connection between the PERMANENT EXISTENCE of an epsilon and the collection of infinitely many elements that is related to it.

 

In other words, there is no way to calculate the exact SUM of infinitely many elements, because the SUM of infinitely many elements cannot be more than SUM – epsilon, and therefore the accurate SUM of infinitely many elements does not exist.

 

Therefore 3.14... < The accurate value of circumference/diameter.

 

 

 

About |N|:

 

The idea of Epsilon = An invariant proportion, is not limited only to a collection that can be found on infinitely many different scale levels.

 

In other words, we can use this idea in order to show that the accurate value of |N| is undefined by definition, where the definition is not else then the ZF Axiom of Infinity, for example:

 

,     ,     ,     ,     ,
|\    |\    |\    |\    |\
| \   | \   | \   | \   | \
|  \  |  \  |  \  |  \  |  \
| [i][b]1[/b][/i] \ | [i][b]2[/b][/i] \ | [i][b]3[/b][/i] \ |...\ | [i][b]n[/b][/i] \  [i][b]n[/b][/i]+1
|____\|____\|____\|____\|____\ ... ad infinitum.

In this case Epsilon = 1, but then we can clearly see the mistake of Cantor's approach, because if n+Epsilon is in N (by the ZF Axiom of Infinity), then the accurate value of N is undefined because we have a permanent state of |N| - Epsilon.

 

 

What to you think?

[matt grime]

Doron, you grace us with your presence, and inane ramblings, again.

 

Glad to see you've not learned anything about mathematics in your absence.

[bloodhound]

hmm. this definitely reads like a doron post. complete with colours as well.

[matt grime]

Oh, it is, lama was one of his aliases in another forum that got banned.

[NSX]

What is an invariant proportion?

 

I can't be bothered to read up on google what it is

[lama]

Dear Matt,

 

After your warm wellcome, please reply a detailed answer about post #1.

 

 

Thank you.

[Sayonara]

Multiple accounts are not allowed.

 

Pick one.

[lama]

I pick lama.

[Sayonara]

Righto.

[lama]

What is an invariant proportion?

Please think about circumference/diameter ratio, which does not depend on a circle's size.

 

The same holds for the infinitely many triangles in post #1.

[ydoaPs]

Please think about circumference/diameter ratio' date=' which does not depend on a circle's size.

 

The same holds for the infinitely many triangles in post #1.[/quote']

 

there are circumstances when the [math]\pi[/math] is not constant.

[matt grime]

ok, here is the only reply you'll get:

 

1. what has the fact that you can draw pictures got to do with provin anything about taking a limit, and then proving that something in the limit at infinity makes any sense?

 

2. who says we calculate the sum of infinitely many things? ONce more you're confiusing applying mathematical operations with a definition of a limit.

 

3. the cardinality of the natural numbers is not a natural number, so there is no reason to suppose it behaves as one

 

3. aleph-0 isn't a "value" it is an equivalence class of sets

 

4. yo'uve failed to define an arithmetic for cardinal numbers so your last bit about addin epsilon, an invariant proportion, to a cardinal amkes no sense

 

5. you've not actually defined this invariant proportion, merely offered an example, and nto said why this is a cardinal number that can be "added" to aleph-0

 

6. do you even know the definition of cardinal numbers?

 

7. cantors definition of cardinals is about bijections of (infinite) sets, so why don't you write a mathematical argument as to why it;s flawed other than saying that infinite sets can't be complete, which is not a mathematical statement, by the way.

 

8 why don't you actually look up the definitions of terms you use so you don't make mistakes?

 

9. why do you think that everything must behave as you wish rather than actually proving what is true?

[zaphod]

this a ****ed up thread.

[lama]

ok' date=' here is the only reply you'll get:

[/quote']

What do you mean by this?

1. what has the fact that you can draw pictures got to do with provin anything about taking a limit' date=' and then proving that something in the limit at infinity makes any sense?

[/quote']

1. there is nothing in the pictures exactly that there is nothing in some agreed formal notations, all is in our mind in this case, and you can understand things not because of a spesific way of representation.

 

If you lost your ability to think abstract thoughts that are not depends on some agreed representation, then I cannot help you.

 

More to the point, in my abstract and precise model I show that there is an inseparable connection between the PERMANENT EXISTENCE of an epsilon that is based on an invariant proportion, and the collection of infinitely many elements that is related to it.

 

This model is too simple to be not immediately understood, it can make hard time only to persons that their abstraction abilities are limited to the agreed formal linear way of written notations.

2. who says we calculate the sum of infinitely many things?

If ALL n is in N then we cannot avoid the idea of a SUM when we think about |N|.

3. the cardinality of the natural numbers is not a natural number' date='...

[/quote']

3. Yes I know, and this is exactly the reason that no natural number is the SUM of all Natural numbers, according to Cantor's approach, which forces a universal quantification, on a collection of infinitely many elements, which is a fundamental mistake, as I clearly show in my abstract model.

4. yo'uve failed to define an arithmetic for cardinal numbers

Wrong' date=' I improved the arithmetic between infinitely many elements, for example:

 

By Cantor Aleph0 = |[b']N[/b]|, which is the cardinality of N set.

 

By Cantor: aleph0+1=aleph0, aleph0-2^aleph0 has no meaning, aleph0 < 2^aleph0, 3^aleph0=2^aleph0, etc...

 

 

My solution to Aleph0 concept

 

My concept of aleph0 is based on "cloud-like" magnitude of any collection of infinitely many elements.

 

For example:

 

aleph0+1 > aleph0

 

If A = aleph0 and B = aleph0 - 2^aleph0, then A > B by 2^aleph0, where both A and B are collections of infinitely many elements.

 

Also 3^aleph0 > 2^aleph0 > aleph0 > aleph0 - 1, etc...

 

Fore more details please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

 

Strictly speaking, Actual infinity (infinitely long non-composed element) is too strong to be used as an input.

 

Potential infinity (infinitely many elements, which never reaches Actual infinity, and therefore cannot be completed) is the name of the game. For further information please look at:

 

http://www.geocities.com/complementarytheory/ed.pdf

 

http://www.geocities.com/complementarytheory/9999.pdf

 

http://www.geocities.com/complementarytheory/Anyx.pdf

5. you've not actually defined this invariant proportion

5. All you have is to look at my representation' date=' and by using its abstract model in your mind you immediately define the invariant proportion, unless you have no ability to think abstract thoughts.

6. do you even know the definition of cardinal numbers?

6. Yes (http://mathworld.wolfram.com/CardinalNumber.html) and their extension to transfinite cardinals can be done only if we force a universal quantification on a collection of infinitely many elements, which is something that cannot be done, and I clearly and very simply show it in my abstract model.

7. cantors definition of cardinals is about bijections of...

7. This bijection cannot be found between collections of infinitely many elements' date=' because endless mapping is a meaningless thing.

 

More to the point: a collection of infinitely many elements cannot be completed, as I clearly show in post #1.

8 why don't you actually look up the definitions of terms you use so you don't make mistakes?

8. Thank you for your advice Matt, but I already did it, and the results of what I discovered are clearly and simply shown in my abstract model at post #1

9. why do you think that everything must behave as you wish rather than actually proving what is true?

9. Ho, dear Matt, no one of us really Knows the true, and therefore why do you think that everything must behave as you wish?

[bloodhound]

 

7. This bijection cannot be found between collections of infinitely many elements' date=' because endless mapping is a meaningless thing.

 

[/quote']

Hmmmm.... that goes againt everything i haven learned in my 1 and a half years of mathematics.

 

you obviously can find a bijection between natural numbers and the rationals

 

thats how countably infinite is defined.

 

or f(x)=x defines a simple bijection from R to R

[matt grime]

I don't think this breaks my promise to respond only once to Doron as it is a reply to bloodhound. (and i must resist the temptation to pooint out his mathematical errors again).

 

Bloodhound, Doron doesn't actually know what a bijection is, or any of the mathematical objects he may use. He may point out a wolfram link as if that explains he understands it, but that isn't the case. His is a simple premise, that any statement that includes a "for all" is automatically false for an infinite set. We know that to be nonsense.

 

For instance where does he define 'endless'? We may have a vague notion that helps explain it, but that doesn't define it se we can use it. Just lookat his attempt at defininf invariant proportion.

 

One need only look at his idea that Cantor defined transfinite arithmetic to see that he's completely ignorant - Conway did that, and there is Robinson's variation. Why, for instance is it important to him that we haven't defined aleph-0 -aleph-1? They are not natural numbers, or a ring for that matter, in any obvious non-trivial way. Of course there is a philosophical issue here that is fundamental: Cantor took the axiom of choice as true in order to well order the cardinal numbers. The cardinal numbers are of course too big to be a set, so there's another important delciate philosophical issue.

 

After all aleph-0 is just a symbol, defined bty the use: a set has card aleph-0 if ther is a bijection with N. Very simple.

 

Very hard is the question of whether 2^aleph-0 which is by definition the cardinality of the power set of N has cardinality aleph-1, the smallest uncountable cardinal. In fact it is independent of ZFC.

 

 

But none of what has been written in post 1 makes any coherent sense.

[lama]

Matt,

 

No one can conclude anything by using mapping between collections with infinitely many elements, because these collections are un-bounded and therefore not completed by their very own nature.

 

All what Cantor did is to show a bijection between a collection of finitely many elements, and then he forced a universal quantification on these collections, in order to get by force his required result.

 

But this result is clearly ill-defined because we cannot conclude something about a collection of infinitely many elements by trying to eliminate by force its unbounded nature in order to define our requested results.

 

Cantor’s extension-by-force of cardinality and ordiality simply does not hold water, so Matt you are the one that continue to force your ill-defined conditions on collections of infinitely many elements by forcing on them a universal quantification.

 

By this forcing method you lose your ability to distinguish between a collection of finitely many elements (where a universal quantification can be related to them) and a collection of infinitely many elements (where a universal quantification cannot be related to them) and this is exactly the deep difference between these kinds of collections, that actually gives us the ability to perfectly distinguish between them.

 

If you ignore this deep and simple difference, then your system is based on ill-defined terms.

 

Simple as that, so do not sell us stories about your "well-defined" universe.

 

Also please pay attention to the very important fact, which is:

 

I am not a constructivist that says that collections of infinitely many elements do not exist.

 

I clearly say that collections of infinitely many elements do exist and they are deeply and totally different from finite collections, and the first and the most important difference is:

 

We cannot force on them a universal quantification, as I clearly and simply show in my abstract model at post #1.

...aleph-0 -aleph-1? They are not natural numbers

I totally agree with you' date=' but I add that they are ill-defined by Cantor, therefore must not be a part of the Language of Mathematics and its logical reasoning.

 

His is a simple premise, that any statement that includes a "for all" is automatically false for an infinite set. We know that to be nonsense.

Really??

 

In Post #1 I clearly and simply show that I am right!!!

 

If you want to show that I am wrong you have to introduce to all of us your model that clearly explain, by using only infinitely many elements, why you are right and I am wrong.

 

You did not do it, you are just talking about it, and just talking is exactly nothing in this case.

 

Also I have noticed that you look at yourself as some kind of a duke or a king, that shares his wisdom with his worshiped people.

 

So, King Matt:

 

I clime that King Cantor is naked … , please prove me wrong!!!

 

In a collection of infinitely many elements, an Epsilon is an invariant NEXT state (which is an inherent property of any collection of infinitely many alamants) that cannot give us the ability to force a universal quantification on this kind of a collection.

f(x)=x defines a simple bijection from R to R

You can conclude accurate things by using mapping, only between collections with finitely many elements.

 

If you disagree with me, then you have to show how a universal quantification can be related to a collection of infinitely many elements.

 

Can you do this?

 

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

 

Some general point of view:

 

There is more than one school of thought in the world of Mathematics.

 

Please look at: http://en.wikipedia.org/wiki/Philosophy_of_mathematics

[matt grime]

Sod it, one more, eh?

 

"If you disagree with me, then you have to show how a universal quantification can be related to a collection of infinitely many elements."

 

 

 

Define "related", define "completed". It is not up to us to show anything because you've not asked of us anything that is written in or about mathematics as is understood by anyone else. Until you phrase your posts in such a way as someone else can make sense of them then we cannot do anything but point out that with the ordinary meanings to all the words you're talking complete rubbish.

 

For all x in R x^2 is positive. What's wrong with that?

 

 

I'm well aware of the different schools of thought in math, Doron, only I seem to understand them and you appear wilfully ignorant of them.

[lama]

For all x in R x^2 is positive. What's wrong with that?

No' date=' it has to be: For [b']each[/b] x in R x^2 is positive, because I clearly show that if R is a collection of infinitely many elements, then we cannot use a universal quantification on such a collection, because:

 

In a collection of infinitely many elements, an Epsilon is an invariant NEXT state (which is an inherent property of any collection of infinitely many elements) that cannot give us the ability to force a universal quantification on this kind of a collection.

 

Do you really can't see this beautiful state?

 

An inherent and invariant NEXT state prevents the use of a universal quantification, it is simple, and it is right in front of your mind, don't you see it?

 

Actually, a collection of infinitely many elements cannot exist without the permanent existence of this Epsilon, and this is an amazing thing, because it gives us the deep insight that the existence of infinitely many elements actually depends on the existence of a single element as its inherent permanent NEXT state.

 

Also please pay attention that I use the word 'state' and not 'process', in order to clarify that I am talking about a timeless simultaneous NEXT, which is an inherent signature of any collection of infinitely many elements...

 

We can use ALL only when we have a collection of finitely many elements.

I'm well aware of the different schools of thought in math' date=' Doron, only I seem to understand them and you appear wilfully ignorant of them.

[/quote']

Please look at my paper about these schools:

 

http://www.geocities.com/complementarytheory/TheBestOf.pdf

 

Now please give your work about these schools, in order to show to all of us, how do you understand these schools of thoughts.

Define "related"' date=' define "completed".

[/quote']

The simple English meaning to "related".

 

"Completed" means that its end can be found.

 

the ordinary meanings to all the words you're talking complete rubbish.

We do not have to think twice in order to understand that Cantor's approach is a complete rubbish that cannot be exist even as an abstract universe.

[haggy]

Doron, we love your MyWay_XOR_TheHighway attitude

[bloodhound]

lol.

[lama]

Doron' date=' we love your MyWay_XOR_TheHighway attitude

[/quote']

If haggy are several personalities (they say:"we love") then I am glad to hear that they agree with each other.

[lama]

 

Doron' date='

 

First of all, I can pretty much assure you I have an ability for abstract thought. In fact, if this were more of a physics problem, I wouldn't be too concerned with having a formal definition for all of your terms and a proof stringing them together.

 

But as it stands this is mathematics, and what you say that we can "understand things not because of a spesific way of representation." falls through because representations of things are fairly important.

 

As best I can tell you're saying that unless a collection of things has a common ratio of some form, then the collection cannot be said to exist. I define now a set {X} that is the set of all prime numbers. Certainly there are an infinite number of primes, but there is no immediately obvious constant that connects all of them (it cannot be a multiplicative constant by definition, nor is it additive, as 3, 5, 7, 11 are all prime and the constant clearly isn't two).

 

Is this what you're getting at?

[/quote']

 

Dear PhysMachine,

 

At these stage let us limit our dialog to abstract (pure) mathematics.

 

So please take post #1 as an abstract model that is not related to, what we call, reality.

 

By these initial terms, a collection of infinitely many elements has no end conditions, or in other words, if we have a well-ordered collection then it can be considered as a collection of infinitely many elements only if its first or its last element (or both first and last elements) cannot be found.

 

By using a universal quantification (the term 'all') on such a collection we actually forcing an impossible condition where both first and last elements of a collection of infinitely many elements can be found, and this collection is still defined as a collection of infinitely many elements.

 

I say that there is a clear XOR connectivity between any collection of infinitely many elements and any definition that tries to force on it its definable ends.

 

In other words, by my point of view, there is a XOR connectivity between a collection of infinitely many elements and definable ends, or in other words:

 

If A is a collection of infinitely many elements and B is its ends, then C is A XOR B.

 

Since a universal quantification forces on us A AND B, then C does not exist as a collection of infinitely many elements.

I define now a set {X} that is the set of all prime numbers...

Then you use A AND B and get nothing XOR a collection of finitely many elements.

 

In other words' date=' there is no connection between the definition of the Prime numbers, and how many of them can be found in some collection, which means: There can be finite or infinitely many prime numbers, but exactly a one and only one definition of them.

 

So the definition is actually the prime number(s) and not their collection.

 

[b']N[/b] is not the definition of the Natural numbers, but only a collection of them, and the Cntorian |N| is based on an A AND B approach, and therefore it does not hold.

[bloodhound]

 

You can conclude accurate things by using mapping' date=' only between collections with finitely many elements.

 

If you disagree with me, then you have to show how a universal quantification can be related to a collection of infinitely many elements.

 

Can you do this?

 

[/quote']

 

so are you saying

[math]f\colon S\to S[/math]

[math]s\mapsto s[/math] (identity map) is not a bijection?

[lama]

If [math]S[/math] has an accurate definition, then the accurate definition is [math]S[/math] and not any finite or infinitely many products of this definition.

 

So mapping between infinitely many elements is an endless thing that cannot return any accurate result, which in this case is the Identity of [math]S[/math].

 

If you want to find the Identity of [math]S[/math], look at its definition (which does not depend on 'Quantity', 'Size', 'Magnitude', 'ALL' s in S, etc...).

 

But if we say that one of the properties of [math]S[/math] is to be infinite, then [math]s\mapsto s[/math] (identity map) is exactly this endless state, and not a bijection of it.