Epsilon = Invariant Proportion

[lama]

Thanks for cheering me up' date=' you can usually be relied upon for saying something hilariously silly.

 

It bears no relation to mathematics though.

 

Just so we can check, do you now accept that threre are infinite sets with a well ordering which possess first and last elements? (max and min, for any mathematicians who may be reading this) just wondering, cos you keep telling me i'm wrong until you evnetually have to accept that actually i'm not, so how long before you accept that when i explain to you how we use "for all" i'm not trying to trick you or force my opinion onto you, but attempting to explain yet another misapprehension you h ave about mathematics. mind you, as you didn't know what a bijection was yet still felt you could assert things abuot them I doubt it'll be anytime soon.[/quote']

 

Dear Matt,

 

Since you still do not understand me, then here is another example:

 

It is obvious that no cardinal or ordinal of the Natural numbers can be found if 1 cannot be found, because 1 is the atom of the rest of Natural numbers.

 

If we force ALL on a collection of infinitely many 1, it means that the successor (+1 expression) cannot be found in this collection.

 

So the results in this case are:

 

A) Since '+1' expression is not in this collection, then only 1 is in this collection, but then this is not a collection of infinitely many elements.

 

B) If '+1' expression is not in this set but we insist that the infinity concept is related to this set, then this set includes an infinitely long non-composed 1 element, but then we are no longer in a model, which is based on infinitely many elements, or in other words, N set is not a collection of infinitely many elements.

 

Your {1/2, 2/3, 3/4, 4/5, 5/6, ... , 3/2} is irrelevant in the Natural numbers case , because the successor (+x) in your example becomes smaller, and this is not the case in the +1 successor of the Natural numbers.

 

So because N is clearly a non-complete collection of infinitely many elements, then the whole idea of 1-1 mapping (bijection or not) does not hold.

 

So 1-1 mapping between collections with infinitely many elements is a non-sense.

It bears no relation to mathematics though.

The Cantorian universe, Dedekind's cut (see post #46), Mapping between collections of infinitely many elements, these are the concepts that has no relation to Mathematics, because they are based on fundamental conceptual mistakes of misunderstanding of the Infinity concept, and they are force the impossible on a collection of infinitely many elements, as I clearly show in the case of the Natural numbers and also in post #46.

 

Matt, you did not show to us how 'ALL' and '+1' (the successor) do not contradicting each other.

 

If you cannot give such an example, than you have nothing to say about my argument.

 

Do you understand this?

[bloodhound]

 

So because N is clearly a non-complete collection of infinitely many elements

 

how can there be a non - complete collection of infinite elements when you were the one who defined a set to be complete only if did not have infitite number of elements

[lama]

how can there be a non - complete collection of infinite elements when you were the one who defined a set to be complete only if [i']did not have infinite number of elements[/i]

It depends of how we understand the word 'complete'.

 

A set is complete only if it's cardinality can be found.

 

As I clearly showed, |N|-1 means, that the cardinality of set N cannot be found.

 

This is the exact difference between a collection of finitely many elements (where its cardinality can be found, therefore it is a complete collection) and a collection of infinitely many elements (where its cardinality cannot be found, therefore it is an incomplete collection).

 

Let us check another thing, which is the structure of the Natural numbers.

 

As we know, 1 is the indivisible atom of N members, so let us show only the sequence of atoms of N members:

 

{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,…}

 

In order to define an order in such a collection, we use subsets of this infinitely long sequence, for example:

 

{{1}, {{1},1}, {{{1},1},1}, {{{{1},1},1},1}, {{{{{1},1},1},1},1} ,…} = {1, 2, 3, 4, 5, …}

 

But it is clear that basic cardinality of both sets is the same |N|-1

 

We can create infinitely many variations of subsets, for example:

 

{{{1},1}, {{{{1},1},1},1}, {{{{{{1},1},1},1},1},1} ,…} = {2, 4, 6, …}

 

But still it is clear that the basic cardinality (the cardinality of the atoms) of any collection of infinitely many subsets has the same |N|-1 .

 

In other words, the deep cardinality is based only on the infinitely long and non-complete sequence of infinitely many 1 atoms, where 1 is an atom because it is the non-composed element of the universe of Natural numbers, and no Natural number can be found, if this atom cannot be found.

 

Now, we can clearly and simply understand, that {1,2,3,4,5,…} or {2,4,6,…} are no more then different variations of {1,1,1,1,1,1,…} which is the basis of any one of them, and this is the deep reason why they can be mapped with each other, because in the case of

|{1,2,3,…}| and |{2,4,6,…}| we actually compare between {1,1,1,1,1,1,1,…} to itself, where

|{1,1,1,1,1,1,1,…}| is no more then |N|-1 .

 

 

By the way, from this deep understanding |{1,2,3}| NOT= |{2,4,6}| because:

 

|{1, 1,1, 1,1,1}| NOT= |{1,1, 1,1,1,1, 1,1,1,1,1,1}|

 

If we insist that {{1},1} is different from {1} then the reason is first of all based on the difference of the caedinality between |{{1},1}| and |{1}|.

 

So how it can be that (for example) |{ |{1}|, |{{1},1}|, |{{{1},1},1}| }| is equal to

|{ |{{1},1}|, |{{{{1},1},1},1}|, |{{{{{{1},1},1},1},1},1}| }| ?

[haggy]

Blah, blah blah.

 

By the above argument, Doron, you are wrong. Can't you see that "Blah, blah blah." is a perfectly valid mathematical argument in your world?

[lama]

No dear Haggy,

 

Please explain to me this Blah, blah blah.

[bloodhound]

i thought the cardinality of [math]\mathbb{N}[/math] was defined to be aleph nought.

 

"As we know, 1 is the indivisible atom of N members..."

 

that to me makes no sense whatever.

[lama]

i thought the cardinality of [math]\mathbb{N}[/math] was defined to be aleph nought.

 

"As we know' date=' 1 is the indivisible atom of N members..."

 

that to me makes no sense whatever.[/quote']

a) The Cntorian aleph nought is no more then a name to something which is based on a contradiction, as I clearly show in this thread.

 

If you disagree with me, then you have to clearly show why the Cntorian aleph nought is not based on a contradiction, so please do that, and if you cannot do that, than you cannot say that the Cantorian aleph nought is definable.

 

b) It is too simple not to be understood, all you have to do is to say that 1 deos not exist.

 

Now please define 2,3,4,5,... cardinality or ordinality without the existence of 1 (or 'First').

[haggy]

Blah => Blah => You're talking (or typing) tripe just like I am now

[lama]

Blah => Blah => You're talking (or typing) tripe just like I am now

Where are the details of YOUR proof?

[haggy]

In fairy land, much like those "proofs" that you think are correct.

[lama]

In fairy land' date=' much like those "proofs" that you think are correct.

[/quote']

I got your point, you are in fairy land, that's why you cannot find your detailed proof about my work, because you look aound you but my work is not in your fairy land.

[matt grime]

"Matt, you did not show to us how 'ALL' and '+1' (the successor) do not contradicting e

each other"

 

willl you please stop indicating to us that you do not what quantifiers mean. We all get that you don't understand them.

 

"If we force ALL on a collection of infinitely many 1, it means that the successor (+1 expression) cannot be found in this collection."

 

 

This makes no sense as a piece of english, nor as mathematics,

 

"Your {1/2, 2/3, 3/4, 4/5, 5/6, ... , 3/2} is irrelevant in the Natural numbers case , because the successor (+x) in your example becomes smaller, and this is not the case in the +1 successor of the Natural numbers."

 

 

again nothing to do with anything - the metric isn't important, we aren't treating these a topological.

 

"So 1-1 mapping between collections with infinitely many elements is a non-sense."

 

Define a map from N to N by: f(x)=x. it is a 1-1 map between two infinite sets. I haven't used any quantifiers. A function is a subset of NxN, this is the diagonal subset {x,x}

 

 

And I see you#re back to stating, without proof, or motivation that infinite cardinals must have the same properties of finite ones.

Of course, you've not defined cardinals yet.

 

Still, as long as you understand yo'ure not talking about mathematics as anyone else understands it then we're ok.

[bloodhound]

somehow , i doubt doron knows what a mapping is.... i gave the same example before as well. but doron refuses to acknowledge that its a mapping between infinite sets. either his definition of a map is completely different from ours, or he doesnt know what he is talking about.

 

if its the first, could u lets us know what u mean by a map

[matt grime]

Yep, 20 years of refusing to learn what other people mean apparently.

 

INcidentally, I think we can sum up his arguments aganis caridinals in more mathematical terms.

 

the idea appears tyo be that with finite cardinals we may define a larger cardinal as a union of two smaller ones. However is we do this with aleph-0 we still only get aleph-0. This isn't a problem, obviously, unless you are of the unswerving opnion that infinite cardinals must behave like finite ones (oddly doron is the one who frequently accuses us of being inflexible, when it is his opinion that wo'nt alter). There are fininte cardinals that are the union over an index of smaller cardinality of smaller cardinals. these have been studied, and we have the term regular because of it.

 

 

if doron wishes to show our definition of an injection from N to N is not valid all he must do is show the negation is true. the negation is that there are distinct naturals x, and y such that f(x)=f(y). as that implies x=y contradicting the assupmtion of them being distinct it could prove difficult. however we can simply expect his disproof to be a flat out denial without evidence.

 

INcidentally, doron has never quite seemed to understand the 3 different levels of criciticism i ahve of his "maths"

 

1. Whatever else, he doesn't make it clear enough until pressed, and even then grudgungly, that he isn't talking about mathematics and using the terms correctly as any one else would understand them. All mathematical statements are imlpicitly made in some system (ie one wher for all has a definite meaning). You may offer new systems for study, offer new definitions, and as long as you are consistent in doing so, no one minds. you may not offer a new definition of some term, and then state that an old theorem is wrong with that new definition as you have changed the hypotheses. this is his error on many occasions.

 

2. this is one that i do not often use. doron has rejected various premises of mathematics. that is not unreasonable. i happen to think his objections are silly, and his new axioms are too, but that is neither here nor there.

 

3. if we are looking at a new system of doronic maths then surely it would behove the author to make clear unambiguous statements about it such as defining ANY of the terms he uses. i do not feel he as done any of this.

 

 

there is also the 4th criticism perhaps that he misrepresents/misunderstands mathematics. he has for instance completely changed his opinions on the ntural numbers - originally he thought they were defined in the axioms of ZFC - they aren't - now he is moving for a set theoretic free idea of peano's axioms. in the face of such shifting it has hard to take seriously any assertions made about extant mathematics.

[lama]

Matt and Dankomed,

 

In the basis (the first-order level) of N set there exists the sequence of infinitely many 1 objects, for example:

 

{1,1,1,1,1,1,1,1,1,1,1,1,1,…}

 

Each Natural number is based on nested finite subsets of this sequence, for example:

 

1= {1}

2= {{1},1}

3= {{{1},1},1}

…

 

and {1, 2, 3, … } = { {1}, {{1},1}, {{{1},1},1}, … }

 

 

The meaning of the Successor is actually, to add 1 to {1,1,1,1,1,1,1, …} sequence, for example:

 

1 = {1} --> |{1}|

2 = {1}+1 = {{1},1} --> |{1,1}|

3 = {{1},1}+1 = {{{1},1},1} --> |{1,1,1}|

…

 

Now by using a Universal quantifier (the term ALL) on N ( where N is a collection of infinitely many finite objects) then if ALL 1 objects are in N, then it implies that +1 (the Successor) cannot be found.

 

Form this result we can conclude that if N is based on infinitely many 1 objects (where each 1 object is a finite and non-composed element), then the logic connectivity between the ‘Universal quantifier’ concept and the ‘Successor’ concept cannot be but a XOR logical connective, for example:

 

(‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) XOR (‘Successor’) --> True

 

But:

 

(‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> False

 

 

 

 

 

If you disagree with me, then you have to prove that:

 

(‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True

 

 

 

Can you prove that?

[matt grime]

"Now by using a Universal quantifier (the term ALL) on N ( where N is a collection of infinitely many finite objects) then if ALL 1 objects are in N, then it implies that +1 (the Successor) cannot be found."

 

This isn#t what we mean when say P(X) is true for all X in N, Doron. Stop being silly.

 

 

"Can you prove that?"

 

AS what yo've jsut asked us to prove is nonsensical statement then of course we can't.

 

I can prove that the function f(x) =x for all x in N is a bijection:

 

Let y be in N and let x=y, then f(x)=y. wow.

 

If x and y arte in N and f(x)=f(y) then by definition x=y.

 

Thus it is a bijection and it's an infinite set and I've quantified something with a for all. there, that wasn't so hard was it?

[lama]

I can prove that the function f(x) =x for all x in N is a bijection:

No you cannot' date=' becuse you did not show how the cardinality of [b']N[/b] can be found.

This isn#t what we mean when say P(X) is true for all X in N' date=' Doron. Stop being silly.

[/quote']

So please explain us what do you mean when you use all X in N where all=Universal Quantifier, where the word Quantifier is actually the word 'Quantity'.

Thus it is a bijection and it's an infinite set and I've quantified something with a for all. there' date=' that wasn't so hard was it?

[/quote']

"My garden flourish in the morning under the midnight Sun" that wasn't so hard to write it, was it?

 

Matt, do you really think that if you can write fancy names and fancy notations, then they are also logically hold?

 

So, I am still waiting to your proof, which clearly shows that:

 

(‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True

[matt grime]

erm, cardinality has nothing to do with it, Doron. i can define functions, injections and bijections without needing to mention cardinal numbers, in fact i must do since i need to define them before i can define cardinal numbers (sometimes you really do make me laugh, wearily). indeed cardinal numbers are sufficnelty complicated as to mean that we would rather not have to discuss them when we do not need to. So, in what way have i done anything inconsistent with the definition of set, injection, surjection and function?

 

For all means, well, for all, as in "for all n in the natural numbers greater than 9, 2^n > n^3" means that if n is any element in the natural numbers and n is greater than 9, 2^n is greater than n^3, or equivalently, there is no n in the natural numbers greater than 9 that does not have this property. That is all. (you may wnat to try proving that statement, since it is true.

 

 

 

So saying for all n in N, there is a succesor in N states exactly the same as "if n in N then n+1 is in N", it doesn't say something about the properties of |N|, or anything of that nature, it says something about the elements of N. that there is no maximal one.

 

It is a short hand, Doron. We could say that

 

2^9>9^3 and 2^10 > 10^3 and 2^11^11^3 and... or we could say it in one simple sentence.

 

So, are you saying that, if S is any subset of the integers, then the statement: for all s in S, s^2 is strictly non-negative (ie is positive or zero) is true if S is a finite set and undefined if S is an infinite subset? Because that is what you seem to be implying.

 

Or how about: for all G in GRP (the category of groups), then if G is finite the order of eery element in G divides the order of G. If I write a proof of that that doesn't mention how many finite groups there are, does it magically become false in your system when i tell you that there is not a finite number of finite groups.

 

You can wait for that proof all you want, however your proposition makes absoltely no sense in the english language or in mathematics or in logic

[lama]

i can define functions' date=' injections and bijections without needing to mention cardinal numbers,

[/quote']

You cannot define any mapping result, if the cardinality of the explored

set(s) is undefined, so you are talking non-sense in this case.

there is no n in the natural numbers greater than 9 that does not have this property.

In this case you do not use the Universal quantifier' date=' but you speak about some property that cannot be found in [b']each single element [/b] in N that is > 9.

 

But you use the Universal quantifier in order to define the cardinality of N.

 

So' date=' are you saying that, if S is any subset of the integers, then the statement: for all s in S, s^2 is strictly non-negative (ie is positive or zero) is true if S is a finite set and undefined if S is an infinite subset? Because that is what you seem to be implying.

[/quote']

Since the cardinality of infinitely many elements cannot be found, then we use: for each s in S, which is good for both finite and non-finite collections.

 

In order to use the word ‘ALL’ you first have to prove that the cardinality of infinitely many elements CAN BE FOUND.

...it doesn't say something about the properties of |N|...

In order to speak about the properties of |N| you first have to define |N|.

 

 

 

So' date=' I am still waiting to your proof, which clearly shows that:

 

(‘Universal quantifier’ [b']AND[/b] ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True

 

because without this proof, |N| cannot be found and no function (mapping) result can be found between collections of infinitely many elements.

[matt grime]

Doron, all of what I have written doesn't depend on defining the cardinality of N. I do not need to define cardinality, indeed I cannot define cardinal numbers without first defining function, injection and surjection. So, you do not need cardinals to do set theory.

 

I do not need to "find" the cardinality of N, whatever thaat means, to show that f(x)=x is a bijection.

 

"for each" and "for all" are semantically the same in mathematics when we use it about elements of some set.

 

 

So I cannot prove, define, explore, whatever you prefer to say, that f(x)=x^2 is not a surjection from N to N? even though it isn't since there is no integer that squares to 2?

 

"In order to use the word ‘ALL’ you first have to prove that the cardinality of infinitely many elements CAN BE FOUND."

 

no, that isn't how other people use "all" that is your view on it, and it is wrong (in the sense that it is inconsistent with its use in the rest of the world).

 

 

"In order to speak about the properties of |N| you first have to define |N|"

 

which part of the sentence where I explain that I'm not stating any result that has anything to do with |N| do you not understand? I don't need to define cardinals, and I don't care too as their definition and proof that they are consistent with, say, ZFC is actually a very delicate matter.

 

 

Anyway, what on earth does it mean to "find" the cardinality of N?

 

 

"But you use the Universal quantifier in order to define the cardinality of N."

 

Do I? Please write out the definition of aleph-0 that you seem to think that I've offered. I mean the proper and rigorous one. And please feel free to show in what way it is that I've proved anything about it. I merely have explained how one can use a universal quantifier to write something about a property that is true for all n in N, and you still don't appear to have grasped its meaning.

 

I cannot prove that last thing you've asked because what you have written is meaningless.

 

 

Please can someon lock this thread? There is no advance to mathematical knowledge from it and Doron is refusing to even countenance that he may need to learn the proper definitions of the terms he is abusing. It is a waste of everyone's time to leave it open.

[lama]

Doron' date=' all of what I have written doesn't depend on defining the cardinality of N. I do not need to define cardinality, indeed I cannot define cardinal numbers without first defining function, injection and surjection. So, you do not need cardinals to do set theory.

[/quote']

Function is mapping to collections of elements, please show us how can you find if (for example) the result is a bijection or not, if the cardinality of this (or these) collection(s) is undefined?

"for each" and "for all" are semantically the same in mathematics when we use it about elements of some set.

And this is the fundamental conceptual mistake of the modern Language of Mathematics' date=' which in its framework 'ALL' = 'EACH'.

 

'ALL' = 'EACH' is a pure non-sense.

no, that isn't how other people use "all" that is your view on it, and it is wrong (in the sense that it is inconsistent with its use in the rest of the world).

In the rest of the word 'ALL' not= 'EACH', and this non-sense is used only by the community of professional mathematicians.

I mean the proper and rigorous one.

Your definition of aleph0 is based on fundamental logical contradiction because:

 

a) In your community 'ALL' = 'EACH' .

 

b) You have no logical proof that clearly shows that:

 

(‘Universal quantifier’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True

 

and I am still waiting to your rigorous logical proof.

Please can someone lock this thread? There is no advance to mathematical knowledge from it and Doron is refusing to even countenance that he may need to learn the proper definitions of the terms he is abusing. It is a waste of everyone's time to leave it open.

Matt' date=' I think that you have to explain how can it be that In your community 'ALL' = 'EACH'?

 

and how (‘Universal quantifier’ [b']AND[/b] ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True ?

 

Can you answer to these two simple questions?

[haggy]

Doron, it is evident that there are two options here:

 

1) You are severely pedagogically challenged

2) You are incorrect

 

So which of the two is it?

 

As for your simple questions: What a load of garbled rubbish!

[lama]

So which of the two is it?

3) You are correct' date=' there is a new quantifier notated as '[b']@[/b]' and called "For Each", and we have a fundamental Paradigm-Shift in the Language of Mathematics and its Logical reasoning.

 

As for your simple questions: What a load of garbled rubbish!

As for your answer: What an Emptiness!

[Dave]

I've had enough of this utter nonsense, hence I'm locking the thread.

 

Doron, if you continue to post this stuff, I will stop you posting in the Mathematics fora.

 

As I've said before: mistakes are fine; ignorance to those mistakes is not.