Hidden Assemption

[matt grime]

OK, so by your definition "sum" cannot be used for an infinite collection. This is not unusual, but so what? You're using, so far, the words sum, cardinality, size in a way no one else agrees with, in this thread.

 

there are two options, at least,

 

1. everyone else is wrong in some absolute sense, and you are right, the whole of mathematics is ill-founded and based upon wrong ideas and we are all trying to suppress your iconoclastic work

 

or, and i'm leaning towards this one to be honest,

 

2. you know nothing about mathematics, and don't understand the idea of axiomatization. (based on, amongst other things, the fact that f(x)=x can sometimes not be a bijection..)

 

 

you are perfectly entitled to take other axioms, but you are not entitled to say we are wrong about them since there is no absolute truth in these issues.

 

you may say that size is something that cannot be applied to infinte sets, or that sum cannot be used. however, no decent mathematician actually uses either of those terms in the way you imply we do. if we do say size it is an illustration only and if pressed we fall back on the proper definition.

 

now, moderators, can we please lock off this thread too? or shall we go through the while pointless routine whereby a crank (this legally is not libellous in the US by the way) tells mathematicians that he understands mathematics better than they do, and the rebuttals which point out that the allegations are neither good english or mathematics, and whilst you may certainly offer a new sustem of mathematics, you ar in no position ot make judgements about the proper, for want of a better word, mathematics are ignored.

[matt grime]

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

 

Thank you.[/quote']

 

 

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

 

it's cardinality, is probably aleph-0 but as you're not giving anywhere near enough information, who knows.

[VazScep]

By the way Doron, do you realise that every single use of a quantifier in ZF set theory ranges over a domain which has no cardinality, namely, the class of all sets?

[lama]

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 ?

[matt grime]

Oh dear God.

 

DEFINITION avoiding the use of all.

 

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

 

defintion of injection, f is an injection if f(x)=f(y) implies x=y.

 

Define a map from N to N, or ANY set to itself by f(x)=x, this is clearly a function, since it is the relation defined by (x,x), the diagonal subset.

 

I;ve not even used the word all once have i?

 

now, f is a bijection since, give y in N, let x=y, and then f(x)=y so it is a surjection, and f(x)=f(y) imlpies x=y, so it is an injection. thus it is a bijection.

 

It;s the sodding definitions.

[lama]

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|?

[VazScep]

So' date=' how can we define 1-1 and onto about N?

 

Where is the proof here?[/quote']From the foundations of predicate calculus? That will take a good deal of exposition.

 

But then I am not sure your beef is with the universal quantifer at all. In fact, you have not demonstrated you know anything about formal logic. It seems your complaint is with proofs written in intuitive logic and featuring phrases such as "for all...".

 

Informally, 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.

 

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 ?

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

[lama]

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.

[lama]

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|?

[matt grime]

Read it carefully Doron, Cantor noted that

 

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

 

5. Initially we say that two sets gave the same cardinality iff there is a bijection between them. Using this, and using analogy, we say things like N is smaller than R, but this is just stating that there is no bijection between them. These are noty pyhsical objects, they do not have mass, width or breadth, this usage of the word size, or smaller or such is a convention that is a generalization of the finite case. It does not contradict the finite case and is reasonable to most people.

 

 

6. starting with this notion of cardinality we ar led to invent cardinal numbers. This is a generaliztion of the order of a finite set. after all we say that

 

{1,2,3} and {2,3,4} have the same size, three elements, in the finite case, so why should we not generalize this to the infinite case and pick some special representative of the class of all sets that are in bijection with N? we do and call this aleph-0, so when we say |S| is aleph-0, which we shall explain in a second, we mean exactly and no more than there is a bijecton with the natural numbers.

 

 

8 having decided that we can use infinite cardinals, cantor showed that, with the axiom of choice the cardinals are well ordered, hence the aleph-0, aleph-1 and so on representatives of the equivlance classes of sets modulo bijection.

 

9. conway then showed how to put an algebraic structure on them.

 

10 when we say aleph-0+1 = aleph-0 then we are saying nothing more than a statement about the existence of maps between the union fo two sets and another.

 

9 sum, size, number as you understand them for the finite case do not have any association to the infinite case a priori. we have generalized them. consequently there are differences between the behaviour in the infinite and finite cases. this is not a problem.

[matt grime]

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.

 

doron, 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.

[lama]

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.

[VazScep]

Yes.

Okay. Now what about the what you've written in between the apostrophes. The phrases "ALL n members" and "Collection of infinitely many finite elements" are not sentences, let alone propositions. Any even if they were

 

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

 

would be a tautology. Any proposition materially implies any true proposition. This fact is often stated as an axiom of propositional calculus:

 

P -> (Q -> P)

 

Rather than trying your hand at logical symbolism, why don't you try straight forward English.

[lama]

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|?

[matt grime]

doron, i am saying nothing abuot all 1-1 maps, nor am i saying anything abuot checking all of them, or "finding" all of them. i have defined |N|, it is the isomorphism class of N in the category SET.

 

please note that i corrected part 1. in my psot to FINITE, as it should read, rather than INFINITE, which is obviously wrong: we are attempting to define a reasonalbe notion that some may call (mistakenly, in my view) size for infinite sets.

[lama]

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.

[matt grime]

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 far doron, there exists exactly one person who thinks the word quantity is what we're talking about. and he/she hasn't defined it mathematically. would you care to correct that?

 

 

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).

 

 

and what, pray tell do you mean by exact? or cardinal? or ALL or related or exist? your paragraph is once more the work of an illiterate.

 

 

 

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

 

what successor? apart from you (a person of very limited mathematical sophistication it has to be said) who has talked about succesors?

 

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.

 

again, no one has mentioned the word successor in the context of defining cardinals of infinite sets...

 

 

 

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|.

 

 

the successor of N? in what way did Cantor define such a thing? he didn't use the phrase "shared property" either, nor did he talk about "infinitely many N members". he used a the axiom of choice to show that there in we may totally order the infinite cardinals, but what's that to with this argument? not a lot as the ordering isn't defined in terms of the successors of peano's postulates...

 

Epsilon = Invariant Proportion

 

are you about to define those terms....

 

 

About 3.14... = circumference/diameter:

 

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

 

oh, that'd be a no then... let us omit some of the junk...

 

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.

 

ahem, so apart from not defining epsilon properly, and once more talking about the VALUE of|N| what is going on here? nothing apart from the ravings of yet another ignorant crackpot....

 

what on earth do you mean by finding the value of |N|? that is yet another piece of nonsense and doen't even begin to explain why the map from a set S to itself as given by f(x)=x isn't a bijection.

 

 

 

So, go on , Doron, please explain why f(x)=x isn't a bijection from S to S. please, pretty please.....

 

And could you at some point explain to the ignroant masses why defining |N| as the iso class of N in SET isn't sound?

[Ophiolite]

Deep breath.......and relax.

[lama]

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

[matt grime]

"The existence of the next"

 

who on earth is claiming that all sets can be well ordered? (they can if we assumu the axiom of choice, but so what)?

 

 

No one else apart from you is talking about successors. They are not important in the argument.

 

So, Let S be any set, and define f:S-> S by f(x)=x.

Show where, with the proper definition of all the terms, this is wrong.

[lama]

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.

[matt grime]

what do you mean by next then?

 

so are you saying that that function is a bijection?

[lama]

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.

[matt grime]

Is it a bijection? A simple yes or no is required, though a proof would be handy.

[lama]

Matt,

 

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