Epsilon = Invariant Proportion

[matt grime]

|N| means, approximately, the class of all sets that are in bijection with N. There is no issue there apart from your ignorance of the meaning of bijection and refusal to accept that inversal qunatifiers are usable with infinite sets. That isn't a problem with mathematics, but with you. So please, go away and develop a mathematical system without universal quantifiers, in which one cannot talk about functions between infinite sets and leave maths alone - it is, so far, consistent in itself.

[lama]

You do understand that?

Do you understand that?

 

In a collection of infinitely many elements' date=' 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.

 

So the cardinality of N is no more then |N| - Epsilon, which means that (the exact) w is undefined.

[lama]

|N| means, approximately, the class of all sets that are in bijection with N

N is a collection of infinitely many elements that cannot be completed, so you cannot use the term all in this case and also you cannot define a bijection (1-1 and onto) in an incomplete collection.

[matt grime]

"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."

 

 

the words epsilon, invariant, next, are undefined, and the conclusion is mathematically unsound in the proper world of mathematics.

 

Tell, you what, why don't you offer a theorem that we know to be true in mathematics and show it is false within that system. Not within your system where none of the terms are defined properly.

 

Since you are talking about cardinality we can only presume you mean in proper mathematics not in your ill-informed mind. So we can only answer in proper mathematical terms.

[matt grime]

N is a collection of infinitely many elements that cannot be completed, so you cannot use the term all in this case and also you cannot define a bijection (1-1 and onto[/b']) in an incomplete collection.

 

 

This just shows that you do not understand what the word "all" means in mathematics, that's, erm, all.

 

If you don#'t like it then use its negation.

[lama]

and leave maths alone

There is no objective thing called Math, and also it is not your privet property, so please hold your horses and be more modest, thank you.

[lama]

"All" mean one and only one thing which is, we can find our first and last elements.

 

We can say it only about finite things.

 

We cannot say it about collections of infinitely many elements.

 

 

This understanding is simple and clear, and any other approach is to define it by force.

 

Edit:

 

A new definition to "all":

 

"All" means one and only one thing which is: No infinitely many elements can be found.

 

 

[Dave]

As much as I'm loving this endless debate, I would like to just step in and say that whilst mathematical discussion is perfectly okay, insults are not. If the thread goes down those lines I'll just close it.

[lama]

Bloodhound and Matt,

 

Please read very carfully post #46.

 

 

Thank you.

[matt grime]

"All" mean one and only one thing which is, we can find our first and last elements.

 

 

That isn't the sense in which we use the word in mathematics. (Definitions are fixed after they have been agreed upon, Doron, you're moving the goal posts again.) This isn't being immodest, it is knowing the accepted meanings of the words.

 

Moreover you are implicitly assuming that all sets are well ordered, and I can show you how to well order every (only) set if we assume the axiom of choice is true.

 

This also highlights what I'm trying to get across to you (and any interested onlookers): you are perfectly free to develop whatever theory you want, and prove whatever results are consistent with it, however you cannot say:

 

Cantor is rubbish because he uses "all" for an infinite set.

 

When you are using, as we can clearly see you have stated, the word in a completely different sense.

 

At best Cantor is "not true" in your system, because you have picked a different system; it is on surprise that one has different results if one changes the hypotheses. Cantor didn't make his statements about a system such as yours, that is all. Stop. moving the goal posts.

[lama]

[math]\{1/2' date='2/3,4/5,5/6,...\} \cup \{3/2\}[/math']

Matt thank you for this, so I have to do some corrections in my definitions, so:

 

A set is complete only if it does not include infinitely many elements.

 

 

Moreover you are implicitly assuming that all sets are well ordered

On the contrary' date=' in my system Uncertainty and Redundancy are first order properties, so ZF or Peano's Axioms define only the 0_Rundancy_AND_0_Uncertainty part of it.

 

 

You have to understand Matt, that no one can stop fundamental changes in our civilization, and Math as part of it, is not protected from deep and comprehensive changes, specially in our time, where updated knowledge is available to anyone in the modern world.

 

The academic institutions are no longer the one and only one alternative to understand and develop our cognitive and intellectual skills, and you can find scholars around the world that can contribute their talents to the society in unaccepted and non-conventional ways that can lead traditional frameworks to new frontiers.

 

This is something which is inherent to the evolution of complex systems like us, and if some developments are useful to us, then they will be spread and will be used by us.

 

Sometimes, there are ideas which come before their time, and most people do not understand them, but if they are fruitful ideas, they will be understood and be developed by our civilization.

 

One of the most important properties of complex systems like us is our abilities to go beyond our current limitations in order to create/discover now frontiers that will be used by us in the near and maybe far future.

 

This ability to act in both tactical and strategic levels are essential to our own existence as complex, yet simple living creatures.

 

If to speak more to the point, I think that I have new insights about fundamental concepts, which are not exclusively belong to what is agreed as the Language of Mathematics, and even today these concepts are understood differently by different schools of thoughts.

 

My goal is to check the possibility to integrate these points of view to a one comprehensive framework, which is based on organic and non-destructive associations between different points of view.

 

By this goal we maybe can re-define and develop the deep sources of life phenomena itself, where the Language of Mathematics and it logical reasoning is one of the most important tools for this goal.

Definitions are fixed after they have been agreed upon ...

This is a wishful thinking of a lot of people, but nothing is totally protected from fundamental changes in our non-trivial reality.

Cantor is rubbish because he uses "all" for an infinite set.

I do not use words like rubbish about anyone's work, but I can explain why I think that Cantor made, in my opinion, conceptual mistakes about basic concepts like Infinity, for example.

 

And I am not just say it, but also explain it in my non-conventional way and also show alternatives to these conceptual mistakes.

 

I do not force anyone to agree with me, but I definitely air my view clearly and do the best I can in order to share my non-conventional ideas with others.

 

The nature of fundamental non-conventional ideas is not to be expressed by the conventional ways, but this is exactly the very nature of fundamental non-conventional ideas, and this is a very hard work to develop a new terminology in order to share these ideas with others.

 

And it is very important to share them with others, because the heart of this thing is to be able to develop a teamwork, which is based on these ideas, and work together in order to develop them, by using an opened mind.

[matt grime]

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.

 

 

Here we are, page 1 of this very thread.

 

Doron, you may work with a differenet definition for the words, perhaps I misspoke to say you shouldn't: there are obviously different meanings for the same word dependent upong context even in maths. However, if you are going to use a different and contradictory meaning then you must clearly say so as the first thing you define, so as not to confuse anyone into thinking that you're talking about their use of the word.

 

At least we can clearly see that you admit you aren't doing mathematics as others do, so we can leave you to your world where "complete" means "finite", which means "first and last element", in which sets have "next elements" but are'nt well ordered apparently, and which contains all other misuses of words.

 

Do you remember the analogy I gave of soemone going to France and refusing to speak French because they thought the meaning of words should be open to change by anyone? Well, it's not a great analogy because words do change their meanine, but not because one person wanders into a foreign country and tells them they're all wrong.

[bloodhound]

Matt thank you for this' date=' so I have to do some corrections in my definitions, so:

 

A set is complete only if it does not include infinitely many elements.

 

 

 

like i said in the other thread, consistency doesnt seem to be ur forte.

 

U can't just go around changing your own definitions at ur whim

 

[edit] what i mean is that you dont just go around changing definitions just to make them fit in ur "theorem". You must set definitions and axioms, and then build a structure around it[/edit]

[lama]

Not exactly dear Bloodhound,

 

I can do that if the new definition is still consistent with the rest of my framework.

 

And it is consistent with the rest of my framework, and also gives a solution to Matt Grime's example.

 

This is a good example of how I can correct my framework through a fruitful dialog with professional mathematicians like Matt Grime.

 

I do not expose my ideas to show that I am "the smartest person in the world" but exactly the opposite.

 

I expose my ideas in order to be corrected by other persons, because, in my opinion, this kind of framework can be developed only through a teamwork.

 

If you find that my correction is inconsistent with my framework, then please reply exactly why this correction is inconsistent with my framework.

 

Thank you.

 

Yours,

 

Doron

[lama]

Dear Matt,

 

I gave a very simple example (the Epsilon>0 argument) of why we cannot use the Universal Quantification with sets like N set.

 

The transfinite universe can be found only if |N| (where N is a well-ordered set) can be found.

 

I clearly show that the best we can get is |N| - Epsilon, which prevents from us to find |N|.

 

So forget about irrelevant examples of non-ordered sets because we are talking here about N, which is a well-ordered set.

 

Now, when you and I sticking only to these initial terms, please show why my |N| - Epsilon does not hold (and my argument is written in a perfect French, unless you prove me wrong in this case).

 

If you are going to use the Axiom of Choice on an unordered N, then it is not relevant, because N was born ordered by Peano's axioms or ZF axiom of infinity.

 

Thank you.

 

Yours,

 

Doron.

[zaphod]

big words + complicated phrasing [math]\not=[/math] mathematical logic

[matt grime]

Please, DOron, attempt to define well ordered (for N) without using a universal quantifier (for an infinite set), and if you wouldn't mind posting, say, Peano's axioms with proof that they are not universally quantified too (for instance, the induction axiom is universally quantified).

 

And your example of N and epsilon is still nonsense - you've not even defined epsilon to the point where anyone knows what you're talking about. and as another example you state that we cannot calculate the sum of infinitely many elements. No one has caclulated any such (algebraic we presume?) sum. What sum?

[lama]

attempt to define well ordered (for N) without using a universal quantifier

1) 1 is a Natural number.

 

2) n = 1

 

3) N is a set of only Natural numbers.

 

4) If n is in N' date=' then [i']n[/i]+1 is in N

 

N is at least a well-ordered set, by the above propositions.

 

Now please show us where is the Universal Quantification here?

[matt grime]

4) can also be written: for all n in N, there is an n+1, a successor in N

 

that's just another way of writing it.

 

can you demonstrate that you must use universal quantifiers in Cantor's argument, as that is you "issue" with it. and thus if i were to rewrite it without universal qunatifiers, then you'd suddenly accept it as true within your system?

 

of course that is all pointless since cantor's theorems are not written in the langauge of your system, whatever that maybe.

 

In cases like this we do not usually say that a theorem is wrong, but that it has no analogue. ie in your doron-maths, there is no analogue of infinite cardinals (incidentally, have yuo defined cardinals yet? don't think you have!)

 

and the other requests to explain what on earth the "infinite" "sum" you seem to think |N| is....?

 

 

incidentally, doesn't the demonstration that there are infinte (well ordered discrete) sets with maximal and minimal elements (something that you said couldn't happen) indicate you ought to tone it down a little? of course it is unclear whether you meant your statement as one about mathematics or your peculiar doron-maths.

[lama]

4) can also be written: for all n in N' date=' there is an [i']n[/i]+1, a successor in N

Matt, in order to re-search fundamental concept like a Universal Quantification you have to return to the moment before Cantor invented |N| and w.

 

At this stage we have: 4) If n is in N, then n+1 is in N

 

Now we start to analyze this proposition.

 

Here is my analysis:

 

If some Natural number is in N than also its successor is in N

 

Since Natural number+1 is also a natural number, then (Natural number+1)+1 is also in N and so is

((Natural number+1)+1)+1 ... ad infinitum.

 

Let Epsilon = 1

 

So the general notation to this idea is (n)+Epsilon.

 

If +Epsilon part of the (n)+Epsilon expression is not in N then N is a finite collection, so in order to get an infinite collection +Epsilon exist in N not as (n) but as the next element after (n) (which is written as +Epsilon).

 

If |N| is the cardinality of all n in N and since (n+Epsilon) is also an N member, then n=n+Epsilon, Or in other words n=n+0, which means that if all n in N then there is no successor anymore because N set is completed by an 'all' term.

 

But if Epsilon=0 because of the all term, and N is also infinite, then we are no longer in a model of infinity that is based on infinitely many elements, but we are in a model that is based on an infinitely long non-composed element, which its cardinality cannot be used as an input by any Mathematical tool.

 

So if we want to keep N as a collection of infinitely many elements, then the cardinality of N (where Epsilon=1) cannot be more than |N| - Epsilon.

 

In order to see my two models of infinity, please look at:

 

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

 

incidentally' date=' doesn't the demonstration that there are infinte (well ordered discrete) sets with maximal and minimal elements (something that you said couldn't happen) indicate you ought to tone it down a little?

[/quote']

Since w cannot exist then {1,2,3, ...} u {w} is acctually {1,2,3,...} u {}.

[Dave]

Let Epsilon = 1

 

*snip*

 

But if Epsilon=0...

 

If you fix epsilon, then you can't go and play around with it like it's variable. Your argument from the "Let epsilon=1" seems to be rather non-sensical, to be the least... n = n + epsilon?!? Don't think so.

[lama]

If you fix epsilon' date=' then you can't go and play around with it like it's variable. Your argument from the "Let epsilon=1" seems to be rather non-sensical, to be the least... [b']n[/b] = n + epsilon?!? Don't think so.

You are right Dave, but since n+Epsilon is also a Natural number, then if ALL n in N (which means that there is no successor anymore) then n+Epsilon=n+0, which is the wrong, but Epsilon is turned to 0 by forcing the term ALL on a collection of infinitely many elements, and I say that we cannot do it, because Epsilon=1 and not 0.

 

If Epsilon is indeed 1 (and not 0), but we also force ALL term on N (ALL n in N)

then we get another impossible result which is: n=n+1.

 

Conclusion: We cannot use the term ALL n in N, or in other words, we cannot force a

Universal Quantification on a collection of Infinitely many elements, like set N.

 

It means that the cardinality of N is |N|-1, and the Cantorian transfinite universe does not exist.

[matt grime]

again you're not using "all" correctly as anyone can see, so what has this to do with mathematics? DOron? In fact god knows how yo'ure using it.

 

the statement "for all n in N, there is an n+1 in N" is exactly the same as the statement "if n in N, then there is n+1 in N"

 

 

if there is a natural number for which its successor is not a natural number, which one is it?

 

are you making statements about mathematics or abotu *your* special doron-maths?

 

and why are you then taking the internal arithmetic of N to conclude things abuot arithmetic of infinite cardinals which you've not defined. that isn't what happens, and I'd hope you'd've learned that by now.

[lama]

if there is a natural number for which its successor is not a natural number' date=' which one is it?

[/quote']

There cannot be a Natural number, which its successor is not a Natural number, therefore Epsilon is a Natural number, and this Natural number is 1.

 

But if we force a Universal quantification on N set, then the successor value cannot be found anymore, because we say that ALL n in N.

 

And ALL n in N has one and only one meaning, which is:

 

The next step or the +1 expression does not exist anymore.

 

In other words: there is a fundamental conceptual contradiction between 'ALL' and '+1' (Successor n) concepts.

 

If you do not agree with me, then please show us that there is no contradiction between 'ALL' and '+1'?

 

Thank you.

 

the statement "for all n in N' date=' there is an n+1 in N" is exactly the same as the statement "if n in N, then there is n+1 in N"

[/quote']

Because 1 is a Natural number and +1 exppression is in N, then N cardinality cannot be more then |N| - 1.

 

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 an Epsilon>0, 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...

 

Another example:

 

The atom of any Natural number is the Natural number 1.

 

If 1 cannot be a successor in a collection of infinitely many elements (because of the ALL term), it means that the Natural number 1 is not in N, and N={}.

[matt grime]

Thanks for cheering me up, 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.