Limit and Infinity

[Guest Doron Shadmi]

Some dialog:

 

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Doron:

 

I think that we do not understand each other.

 

I gave you MY definiton of the limit concept.

 

Now, please give the standard definition for this concept.

 

After you give the standard definition, then we shall compare between

the two approaches.

 

Any way do you agree with http://mathworld.wolfram.com/Limit.html definition?

 

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kaiser:

 

off course I agree with this definition. I meant for you to provide the defintion for the limit of S(n), no need delta epsilon at this point. A limit can be defined using epsilon and S(n). At any case, I am not interested in your definitions at the moment. I need to be convinced that you understand and know how to use the fundamental "conventional" mathematical defintions before we can move on to your definitions.

 

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Doron:

 

Ok, the main persons in modern Math that are related to the so called rigorous definition of the limit concept are Cauchy and Weierstrass.

 

Cauchy said:" When some sequence of values that are related one after the other to the same variable, are approaching to some constant, in such a way that they will be distinguished from this constant in any arbitrary smaller sizes that are chosen by us, then we can say that this constant is the limit of these infinitely many values that approaching to it."

 

Weierstrass took this informal definition and gave this rigorous arithmetical definition:

 

The sequence S1,S2,S3, … ,Sn, ... is approaching to (limit) S if for any given positive and arbitrary small number (e > 0) we can find a matched place (N) in the sequence, in such a way that the absolute value S-Sn (|S-Sn|) become smaller then any given epsilon, starting from this particular place in the sequence

(|S-Sn| < e for any N < n).

 

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kaiser:

 

Very good! now based on the definition you provided, which is a correct mathematical definition please find out the limit of the following sequence:

 

0.9,0.99,0.999,0.9999,0.99999,....

 

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Doron:

 

-------post #190

 

Now please listen to what I have to say.

 

First please read http://www.geocities.com/complementarytheory/9999.pdf

(which is also related to your question) before we continue.

 

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Doron:

 

-------post #191

 

I disagree with the intuitions of Weierstrass, Cauchy, Dedekind, Cantor and other great mathematicians that developed the current mathematical methods, which are dealing with the Limit and the Infinity concepts.

 

And my reason is this:

 

No collection of infinitely many elements that can be found in infinitely many different scales, can have any link with some given constant, in such a way that it will be considered as a limit of the discussed collection.

 

In short, Nothing is approaching from the collection to the given constant, as can be clearly seen in my sports car analogy at page 2 of http://www.geocities.com/complementarytheory/ed.pdf

 

Take each separate position of the car, then compare it to zero state and you can clearly see that nothing is approaching to zero state.

 

Therefore no such constant can be considered as a limit of the above collection.

 

It means that if the described collection is A and the limit is B, then the connection between A,B cannot be anything but A_XOR_B.

 

So here is again post #184:

 

Since I am not a professional mathematician, my best definition at this stage is:

 

A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.

 

It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.

 

By using the word "leap" we mean that we have a phase transition from state A to state B.

 

There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.

 

A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.

 

From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.

 

If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.

 

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Doron:

 

-------post #192

 

'Any x’ is not ‘All x’

 

 

By inconsistent system we can "prove" what ever we want with no limitations

but then our "proofs" are inconsistent.

 

A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proven by the current axioms of this system, and we need to add more axioms in order to prove these statements.

 

So any consistent system is limited by definition and any inconsistent system is not limited by definition.

 

 

Let us examine the universal quantification 'all'.

 

As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements, even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements is an inconsistent idea.

 

For example:

 

Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.

 

In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.

 

Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf

 

Form this point of view a universal quantification can be related only to a collection of finitely many elements.

 

An example: LIM X---> 0, X*[1/X] = 1

 

In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.

 

'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.

 

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kaiser:

 

If you do not see that the limit of the sequence I provided is 1, then you do not understand what a limit is, and therefore can not agree or disagree with its definition.

 

In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. A sequence does not have a limit, if it is not approaching some constant, for example the sequence 1,2,3,4,... does not have a limit, it disperses to infinity.

 

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Doron:

 

1 as the limit of the sequence 0.9,0.99,0.999,0.9999,0.99999,.... is based on an ill intuition about a collection of infinitely many elements that can be found in infinitely many different scales, as can be clearly understood by posts #190,#191,#192.

 

You can show that 1 is really the limit of sequence 0.9,0.99,0.999,0.9999,0.99999,.... , only if you can prove that there is a smooth link (without "leaps") between this sequence and 1, which is not based on {0.9,0.99,0.999,0.9999,0.99999,.... }_XOR_{1} connection.

 

Maybe this example can help:

 

r is circle’s radius.

 

s' is a dummy variable (http://mathworld.wolfram.com/DummyVariable.html)

 

a) If r=0 then s'=|{}|=0 --> (no circle can be found) = A

 

b) If r>0 then s'=|{r}|=1 --> (a circle can be found) = B

 

The connection between A,B states cannot be but A_XOR_B

 

Also s' = 0 in case (a) and s' = 1 in case (b), can be described as s'=0_XOR_s'=1.

 

You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 --> 1

 

A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

 

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terrabyte:

 

what kaiser is saying is that 1 is the limit, but 1 is not included in the set of .9+.09+.009...

 

i THINK what Doron is saying is:

intuitively the number this "approaches" is 1, getting infinitely close to but never reaching it. but actually the number it really "approaches" is .999...

 

in other words you're both saying the same thing

 

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Doron:

 

...getting infinitely close...

"getting close" is reasonable.

 

"getting infinitely close" is not reasonable, because nothing can be closer to something when something is some constant and the "closer" element is one of infinitely many elements that can be found in infinitely many different scales.

[Guest Doron Shadmi]

I think you left out the most interresting posts. The ones where Kaiser asked you to calculate the limit of 0.9' date='0.99,0.999,0.9999,0.99999,.... using your definition.

[/quote']

 

Dear AndersHermansson,

 

First, Kaizer asked me to use the standard way to show how we calculate the limit of 0.9,0.99,0.999,0.9999,0.99999,.... , after I showed him that I understand the standard interpretations of the Limit concept.

 

But you are missing the goal of my thread.

 

I am not talking about some practical tool, that can be used by us to get some desirable result, which is limited to the current understanding of Limit and Infinity concepts.

 

My goal is to show how the most simple and intuitive concepts of the language of Mathematics can be understood in a different way, which can lead us to develop it in new directions.

 

For example please look at my paper here:

 

http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

Thank you,

 

Doron

[Guest Doron Shadmi]

Hi Gza,

I was just curious Doron: What are the problems with the number system that exists today that would make you want to change it? Like they say' date=' "if it aint broke, don't fix it."

[/quote']

Today's number system is a quantity-only system, which ignores the internal complexity of the natural numbers (and I do not mean to the differences between primes, non-primes, odds , evens, partitions, permutations, etc..., which are all based on 0_redunduncy_AND_0_uncertainty building-blocks), which are the building-blocks of the entire standard system.

 

In short, my number system is based on the information concept, where each building-block in it has an internal structure that cannot fully described only by quantitative-only and 0_redunduncy_AND_0_uncertainty approach of the standard system.

 

The main concept of my new number system is based on the complementary relations that exist between symmetry level and information's clarity-level, and these relations are based on what I call complementary-logic, which is based on an included-middle reasoning, and both excluded-middle reasoning and fuzzy logic are limited proper sub-systems of it.

 

By my system we get these benefits:

 

1) Each building-block has a unique internal complexity, that can be the basis for infinitely many unique building-blocks, which can be found upon infinitely many different scales.

 

2) There are infinitely many unique internal structures that can be found in some particular scale level.

 

3) There can be infinitely many complex structures, that are based on (1) and (2) building-blocks combinations.

 

4) These complex structures are much more accurate models then any model which is based on the quantitative-only standard number system, and some of the reasons are:

 

a) The structure that is based on the complementary relations between symmetry and information concepts (where redundncy_AND_uncertainty are useful properties of them) is inherent property of my new system, and gives it the ability to understand the deepest principles of any dynamic/structural abstract or non-abstract complex object, without first reducing it to a quantitative model (which is inevitable when we use the standard quantitative-only number system).

 

b) The new natural numbers (which are now taken as topological information's building-blocks) are ordered as Mendeliev-like table, which gives us the ability to define their deep topological connections, even before we use them in some particular model.

 

These deep topological connections can be used as gateways between so-called different models, and expending our understanding about these explored models.

 

c) My number system is the first number system, which is based on our cognition’s ability to count, as an inherent property of the abstract concept of a Number.

 

By this research I have found and described how the number concept is based on the interactions between our memory and some abstract or non-abstract elements.

 

Through this approach our own cognition is included in the development of the Language of Mathematics, and we are no longer observers, but full participators where our own congenital abilities are legitimate parts of the mathematical research itself.

 

For example:

 

What is called a function is first of all a reflection of our memory on the explored elements.

 

A function is the property that gives us the ability to compare things and get conclusions that are based on this comparison.

 

If something is compared by us to itself, we get the self identity of an element to itself by tautology (x=x).

 

If more then one element is compared, then we get several information clarity degrees that describe several possible interactions between our memory and the explored objects, and these several possible interactions can be ordered by their internal symmetrical degrees.

 

In this case multiplication and addition operations are complementary operations, where multiplication can be operated only between identical elements (redundancy_AND_uncertainty > 0) and addition is operated between non-identical elements (redundancy_AND_uncertainty = 0).

 

Because any function (which is not based on self reference of an element to itself) is a connection between at least two elements, its minimal abstract model cannot be less then a pointless line-segment, which is used as a connector between the examined elements.

 

In this case no interval (memory) can be described in terms of points (objects) and vise versa, and we get these four independent building-blocks of the language of Mathematics (which now includes the mathematician’s cognition-abilities as a legitimate part of it):

 

{}, {.}, {._.}, {__}

 

By this new approach we can build, for example, a totally new Turing-like machine, that can change forever our abilities to deal with complexity which is based in simplicity.

 

Please look at my website http://www.geocities.com/complementarytheory/CATpage.html if you want to understand more.

 

 

So, if we return to your first question, is this a wise thing to get off the evolution process?

[Guest Doron Shadmi]

Doron; what do you propose we use if we want to talk about quantity?

 

Then continue to use only 0_redundancy_AND_0_uncertainty building-block.

 

And if you want to avoid any change of the current number system' date=' then ignore the duality of any [b']R[/b] member, which can be seen in http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

Also continue to use universal quantification as a deductive concept that can be related to a collection of infinitely many elements.

 

Also ignore memory/object(s) interactions, as a fundamental must-have condition that standing in the basis of the Number concept.

 

Also ignore Symmetry/Information complementary relations.

 

Also ignore {__} (the full-set) which is the opposite of {} (the empty-set).

 

Also ignore {._.} building-block and continue to use only {.} building-block.

 

Also ignore Multiplication/Addition complementary relations.

 

Also ignore Complementary-logic and continue to use only Excluded-middle reasoning.

 

In short, avoid any possibility of evolution process in the Language of Mathematics.

[JaKiri]

Something happened here, I'm not sure what

[Guest Doron Shadmi]

Hi JaKiri,

 

You have a beautiful avatar.

[bloodhound]

I would like to add my opinions. correct me if i am wrong

 

Some dialog:

 

A Limit is any arbitrary well-defined element' date=' where no collection of well-defined infinitely many elements can reach it.

 

It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.

 

By using the word "leap" we mean that we have a phase transition from state A to state B.

 

There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.

 

[/quote']]

But doesnt the sequence {0,0,0,0,0,0,0,0,0,0,...} have a limit 0? Can you explain what you mean by phase transition. This thread is way over my head.

[Guest Doron Shadmi]

But doesnt the sequence {0' date='0,0,0,0,0,0,0,0,0,...} have a limit 0? Can you explain what you mean by phase transition. This thread is way over my head.

[/quote']

By using the words 'phase transition' I mean that at least two different states have a XOR connection between them, or in other words, there is no smooth link between them.

 

{0,0,0,0,0,0,0,0,0,0,...} is {0} by "normal" set definition (http://mathworld.wolfram.com/Set.html) an independent set if it is considered as a multi set (http://mathworld.wolfram.com/Multiset.html), but then {0} and {0,0,0,0,0,0,0,0,0,0,...} are two independent mathematical objects.

[bloodhound]

i am not talking about a set, i am talking about a sequence of 0's

 

i still dont know what u mean by phase transition. or by XOR connection, or smooth link. you use them to describe each other and i have no idea what ur on about.

[Guest Doron Shadmi]

please explain how {0,0,0,0,0,...} and the limit 0 are mathmatically connected to each other.

[bloodhound]

oops maybe clash of notation

 

take sequence [math](x_n)[/math] where [math]x_{n}=0[/math] for all [math]n \ge 1[/math]

 

then the limit of that sequence is obviously 0

[ed84c]

anybody like to teach me calculus? Ive been searching for a teacher for ages. (no im not taking the piss im just sad)

[Guest Doron Shadmi]

I disagree with the deductive universal quantification approach, and I clealry explain why a deductive universal quantification cannot be related to a collaction of inifinly many elements.

 

Please read my first posts and see by yourself.

 

I think that your example is not connected to the standard mathematical interpretation of the limit concept (http://mathworld.wolfram.com/Limit.html).

 

Please explain why do you call 0 the limit of the sequence {0,0,0,0,0,...}

[ed84c]

i also have a problem regarding calculus; the differential quotient states to make the limit of[delta]x 0 and then over that you put some other equation i cant remember. But if [delta]x has a limit of 0, doesnt that mean divinding this out will always give infinity? :S

[bloodhound]

from what i have learned till now at uni

 

a real sequence [math](x_n)[/math] is said have a limit [math]L \in \mathbb{R}[/math]

if the following is true. To each positive number a, corresponds an integer [math]n_0[/math], such that [math]|x_n-L|[/math] is less than a for all integers [math]n \ge n_0[/math]

 

obviously this condition satisfies if the sequence is 0,0,0,0,0,... and L is taken to be 0, for all n.

 

Also we can use the monotonic sequence theorem, which states that

 

let [math](x_n)[/math] be a sequence which is non decreasing for [math]n \ge N[/math]. If [math](x_n)[/math] is bounded above, then (x_n) converges, and the limit is the supremum of the s of the set [math]{x_{n}:n \in \mathbb{Z},n \ge N}[/math]. If (x_n) is not bounded above, then (x_n) diverges to [math]\infty[/math]

 

so from above the limit of sequence 0,0,0,0,0,.. is 0, as the sequence is non decreasing. and 0 is the lowest upper bound of the set {x_n}

 

the same theorem can also (but pointless) be applied to show that the limit of the sequence

 

(0.9,0.99,0.999,0.9999,...) is 1.

[NSX]

i also have a problem regarding calculus; the differential quotient states to make the limit of[delta]x 0 and then over that you put some other equation i cant remember. But if [delta']x has a limit of 0, doesnt that mean divinding this out will always give infinity? :S

 

 

What is a differential quotient?

[ed84c]

sorry the difference quotient

[Guest Doron Shadmi]

Dear bloodhound,

 

This is the whole point in this thread.

 

I disagree with the standard interpretations of the Limit and the Infinity concepts.

 

So please read step by step all of my first posts (including their links) and then please give your response.

 

Thank you,

 

Doron

[Guest Doron Shadmi]

Dear ed84c and NSX,

 

This thread is dedicated to a non-standard interpretation of the Limit and the Infinity mathematical concepts.

 

So, it will not help you to get standard answers to your questions.

 

Yours,

 

Doron

[bloodhound]

oh well. sorry. the threads to long for anyone to properly read. Maybe you can put up a summary of key points at the beginning?

[ed84c]

my apologies also

[Guest Doron Shadmi]

Dear bloodhound,

 

It is very hard to write a summery of points to a non-standard point of view on fundamental concepts of the language of Mathematics.

 

You first have to "enter" to my head by reading step by step my first posts, and then compare your knowledge with my new points of view.

 

Thank you,

 

Doron

[bloodhound]

Since I am not a professional mathematician' date=' my best definition at this stage is:

 

A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.[/quote']

Like I said the sequence 0,0,0,0,0,0... has the limit 0

It means that if A is the collection of infinitely many elements and B is the limit' date=' then we can reach B only if we leap from A to B and vise versa.

 

By using the word "leap" we mean that we have a phase transition from state A to state B.

 

There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.

 

A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.

 

From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.

 

If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.[/quote']

I still dont know what u mean by "leap", still don't know what u mean by"phase transition", still dont know what u mean by "Smoothly links". U ask us if we disagree, to define a smooth link between A, B states. but this sentece "There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states." you talk as if you already know what a smooth link is defined as.

[Guest Doron Shadmi]

Like I said the sequence 0,0,0,0,0,0... has the limit 0

Why do you think that 0,0,0,0,0,0,... is limited by 0?

I still dont know what u mean by "leap",

If 0,0,0,0,0,0,... reaches 0 then it is not a collection of infinitely many elements anymore.

 

It means that there is a XOR connection beween a collection of infinitely many elements and some given constant.

 

Therefore 0 cannot be considered as a limit for 0,0,0,0,...

 

The transition from being a collection of infiniely many elements to a collection of finitely many elements that reaches 0, is a "leap".

 

In short, there is not smooth link (without any leap) that can keep 0,0,0,0,... as a collection of infinitly many elements that also reaches 0.

[bloodhound]

anything can be shown by changing the definition of something.

 

using the normal definition of a limit

 

0,0,0,0,0,0...... does have the limit 0.

 

I dont see what ur point is.. What exactly are you trying to show us in this thread? that all the work done by mathematicians since ages ago are flawed?