lama

In the attached page http://www.geocities.com/complementarytheory/no1.pdf we can understand that there is a deep connection between our abstract ideas and the ways that they are represented by us.

I'll be glad to know your point of view.

Cantor's second diagonal method is not from N to R because each element in the list is only a non-accurate representation of R member.

Please open: http://www.geocities.com/complementarytheory/NEXT.pdf

Thank you.

Let us say that your own life depends on your ability to explain the Bijection concept to a 5 years old child.

Please write your explanation.

Thank you.

Cantor's second diagonal method is not form N to P(N) because each element in the list is only a non-accurate representation of R member, for example:

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.

So we can clearly see that any Base # representation of R member is always < R member.

Conclusion:

Cantor’s second diagonal method is not between N and P(N) but between N and N.

In this case we can immediately use the diagonal number in order to proof that the cardinality of a non-finite collection cannot be found because:

The map f(z)=z-(diagonal number) is not a bijection from Z to Z.

You claim that you cannot quantify for all for infinite sets. Yet you're saying' date=' from one example that has nothing to do with the matter in hand that all maps between all infinite sets are not bijections. How can you say that? YOu've jsut qunatified an infinite set with a for all.

[/quote']

If I said it, then it is my mistake.

 

Again:

 

The difference between a finite set and a non-finite set is:

 

A bijection cannot be found in a non-finite set.

 

Proof: Cantor's second-Diagonal method.

Matt please read carefully post #64

Ok, If Matt cannot do it, then let us continue.

Now Matt as I clearly explained, the existence of a non-finite collection depends on the 'existence of the next element' as we can clearly show, by using Cantor's second-diagonal method.

In a non-finite collection, the diagonal number must be permanently added to the collection.

In a finite collection, the diagonal number must not be added to the collection.

We clearly know that we cannot define a bijection in the second diagonal method, because each time we define a bijection, we discover that there is a new element which is out of the domain of our 1-1 correspondence mapping (a permanent next element).

Conclusion:

No bijection can be found in a non-finite collection, and this property can be found in any given non-finite collection and only in a non-finite collection.

Therefore the cardinality of a non-finite collection, like set N (for example) is |N|-NEXT.

A very simple question that needs a very simple answer.

equivalently a map is a beijction between sets if and only if it possesses an inverse.

 

As you've said the bijections are fundamentally wrong (as are cardinals) why am I explaining this to you? surely you know the definition (otherwise' date=' how can you have formed an opinion).

 

nb, the inverse of a set function f from X to Y is a map g from Y to X statisfying

 

gf(x)=x, and fg(y)=y whenever x is an element of X and y is an element of Y.

[/quote']

Matt,

 

Let us say that I am a 5 years old and you want to explain to me what is a bijection.

 

Can you do that?

Bloodhond are you also Matt Grime?

I am still waiting to Matt's very simple English exaplanation of the Bijection concept.

No Blarg,

I am waiting to Matt's explanation of the bijection concept, in a very simple English.

Matt,

Please explain in a very simple English what is a bijection?

so are you saying that that function is a bijection?

Any given collection is identical to itself.

what do you mean by next then?

The existence of a non-finite collection depends on the 'existence of the next element' as we can clearly show, by using Cantor's second-diagonal method.

In a non-finite collection, the diagonal number must be permanently added to the collection.

In a finite collection, the diagonal number must not be added to the collection.

Dear Matt,

The existence of the next has nothing to do with order.

f:S-> S by f(x)=x.

The only notion by these notations is:

Any given collection is identical to itself.

Dear Matt,

Finite collection and non-finite collection cannot be joining together to a one class, as Cantor tried to do.

And the reason is very simple.

In a finite collection the successor>0 is not a permanent property, and because of this reason we can define the Cardinal of a finite collection.

But in a non-finite collection a successor>0 is a permanent property, and because of this reason we cannot define the Cardinal of a non-finite collection.

Actually we can use Cantor’s diagonal method in order to prove it.

The diagonal number which is not in the list, simply proves that we cannot define the complete list of a non-finite collection, because this diagonal number is actually the permanent next element that cannot allowed us to define a complete collection of infinitely many elements.

Cantor did not understand its own diagonal method, because he used the hidden assumption that such a complete list of non-finite collection can exist, by ignoring the fundamental difference that exists between a finite collection and a non-finite collection.

And this fundamental difference is reduced to one and only one property, which is:

The existence of the next.

The permanent existence of the next is a fundamental property of a non-finite collection.

The non-permanent existence of the next is a fundamental property of a finite collection.

Very important:

We cannot define a one class for both of them because there is a XOR connectivity between a finite collection and a non-finite collection that can be written as:

Finite collection XOR Non-finite collection.

Yours,

Doron

i have defined |N|' date=' it is the isomorphism class of N in the category SET

[/quote']

So we see that there can be found some common property of some class (a collection of sets/elements that share some common property), but this common property is not related to the Quantity concept, and therefore it cannot be SUMMERIZED, or in other words, this common property cannot be summarized beyond the quantity 1, and the meaning of quantity 1 in this case is synonym to the term ‘There exists’.

 

So by this common property there exists |N| but it cannot be used as the exact cardinal of all N members, because the term 'ALL' cannot be related to a collection of infinitely many elements, and this collection actually exists because of the existence of a successor > 0 (in the case of N, the successor=1).

 

This Successor is also a common property of class of N, which its permanent existence prevents N from being a finite SET.

 

Therefore the Cardinality of N (which is a non-finite set) is no more than |N|-Successor, or in other words, the cardinality of N is undefined.

 

So as you see Matt, Cantor used only half of the story and defined |N|, but he did not notice that the successor of N is also a shared property of infinitely many N members that permanently prevents from us to define |N|.

 

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.

doron' date=' that is nonsense, utter tripe. look up the definitions. read them, understand them. try getting someone to translate to english for you. i am saying nothing about the "complete collection of 1-1 mapping", at least i don't think i am, since that phrase is meaningless.

[/quote']

But you contradict yourself, because if you check each 1-1 mapping separately then how can you conclude GENERAL CONCLUSION ON ALL OF THEM, IF ALL OF THEM CANNOT BE FOUND?

 

In other words, all you have is the definition, so please tell me, is this definition is actually |N|?

1. we have a clear idea of what we mean by size for infinite sets.

 

2. this could be summarized by : two finite sets have the same size if and only if there is a bijection between them.

 

3. that this equivalent notion does not mention whether or not the sets are finite.

 

4. that we can thus use this GENERALIZATION of the notion of size of finite sets to distinguish between infinite sets in an ANALOGOUS manner.

There is no logical proof here but an extension by using (bad) intuition, as I clearly show in: http://www.geocities.com/complementarytheory/EProp.pdf

Since the rest is based on these 4 paragraphs, then they do not hold.

No idea what this means. Is "-->" implication?

Yes.

Informally' date=' then, f: N -> N can be defined by f(n) = n. This is a bijection, since, whatever n we may choose, n is the image of itself and only itself under f.

[/quote']

Beautiful, now please show how can we conclude by bijection, what is |N|?

F:X --> Y is a surjection if for each y in Y' date=' there is some x in X with f(x)=y

[/quote']

But then your definition is good only to each 1-1 mapping and you cannot extend it to the all complete collection of 1-1 mapping.

 

It means thet |N| is undefined.

what on earth do you mean by number? because you apparently have different definitions from the rest of us.

"Cantor invented the one-to-one correspondence' date=' which easily showed that two finite sets had the same cardinality if there was a one-to-one correspondence between the members of the set. Using this one-to-one correspondence, he transferred the concept to infinite sets; i.e the set of natural numbers N = {1, 2, 3, ...}. He called these cardinal numbers transfinite cardinal numbers, and defined all sets that had a one-to-one correspondence with N to be denumerably infinite sets."

 

http://en.wikipedia.org/wiki/Cardinal_number

[/quote']

So how Cantor defined |N|?

Cardinality is a property of a set defined in terms of mappings' date=' which themselves are defined in terms of individual members of the set.

[/quote']

So, how can we define 1-1 and onto about N?

 

Where is the proof here?

 

Or in other words, please show me how Can you conclude what are thet:

 

(‘ALL n members’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True ?

SUM is the number of elemets that can be found in a finite collection, and the word 'ALL' can be related only to finite collections.

the sentences P is true for all members of S and P is true for each member of S roughly are the same.

I agree with you but only S is a finite collection.

Again:

Please show us stap by stap how can we define the complete number of elements in {1,1,1,1,1,1,1,...}?

Thank you.

I would say that both phrases are intended as a shorthand for "whatever person you choose" or "given a person" etc...

So, as you see you also related to to level of a single member and not to all members at once.

All members at once is meaninful only if their exact SUM also can be found, but in a non-finite collection, we can talk only in the level of a single given member.

A Bijection does not change this state, so how can we find the exact Cardinality of N, using a 1-1 (look for youself a bijection is based on 1-1, or in other words, on the level of a single object) and onto?

Furthermore, what is the meaning of onto and how can we translate it to an exact Cardinal?

well' date=' if you ever bothered to use the Universally acknowledged DEFINITION of a bijection then it should be obvious to you.

[/quote']

I know what is 1-1 and onto, but how you can use it to define the size(s) of collections of infinitely many elements?

 

and please do not give me Cantor's second diagonal, because first show us how you find the exact value of |N|.