Russell's Paradox and the Excluded-Middle reasoning

[kaiser soze]

Lama,

Tell you what, convince me that you understand fundamental mathematical definitions and I will gladly enter into discussion with you about them. Please provide a mathematical definition for issue (b) in post #176.

Kaiser.

 

[Lama]

Definition is not the point in this philosophical level, understanding is the point in this philosophical level.

Please read all of http://www.geocities.com/complementarytheory/ed.pdf
where you can find my new intetpretation to b,c,d subjects of #176.

If you do not like my answer then please define 'definition' and I'll try to give my answer by your definition to 'definition'.

a,b,c,d of #176 can be found in http://www.iidb.org/vbb/showpost.php?p=1716483&postcount=76

 

[kaiser soze]

Lama,

I need to be sure you understand fundamental mathematical definitions as they are understood by mathematicians. You have stated many times before, that one should keep an open mind and develop criticism based on understanding. I am not sure you understand fundamental mathematical concepts, yet you criticize them on a daily basis.

Please provide a mathematical definition to issue (b) of post #176. At this stage your interpretations are irrelevant, I am talking about mathematical defitinions that would be accepted by any mathematician.

Kaiser.

 

[Lama]

Dear kaiser soze,


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.

Please look again on My Riemann's Ball example, in page 3 http://www.geocities.com/complementarytheory/ed.pdf



Thank you.

Lama

 

[kaiser soze]

Lama,

By any standard this is not a mathematical definition of a limit. This is what I mean when I say that you lack the knowledge of fundamental mathematical definitions. A definition of a limit can be found in any calculus textbook - usually this is the subject of the first lecture in undergarduate school.

Kaiser.

 

[Lama]

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?

 

[kaiser soze]

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.

Kaiser.

 

[Lama]

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).

 

[kaiser soze]

Lama,

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,...


Kaiser.

 

[Lama]

Dear kaiser soze,

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.

Thank you.

Lama

 

[Lama]

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.


Thank you.

Lama

 

[Lama]

'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.

 

[kaiser soze]

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.

Kaiser.

 

[moshek]

Since the context here is new mathematics I aloud myself to share with you this problem:Few month ago i found in one of the popular books about Wittgenstein a quote that Wittgenstein describe a possibility to create new mathematics with the geometry of Klein bottle, I am searching now for the exact reference if anyone can help me with that I will thank him


Moshek​

:surprise:

 

[Lama]

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.

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

 

[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 Lama 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

 

[Lama]

...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.

 

[Moscowjade]

I think you are confused about how something can be quantized by NOT, when x = x is a statement of IS. The paradox lies in the perception of non-existence, which by its own definition, cannot exist.

The paradox comes from trying to divide 1 by zero, because zero goes into one zero times. All numbers can be divided by one, so it is irrelevant to suggest that all other numbers can't be divided by zero either. However, we use zero in order to quantize one because without it, you can't quantize infinity. One always equals one. So, zero or NOT is just a tool of perception we use to quantize one, or IS. IS always equals IS just as x always equals x. X does not replicate from x, but when quantized with the perception of _x, x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x ,ad infinitum. The paradox is just a perception of x not being x.

In my mind, zero can't change one, but it can bend your perception of one where
1/0 = C, where C is zero rippling the perception of one into 10101010101010101010101010101010101010101010101010101010101010101010.

 

[Lama]

The paradox lies in the perception of non-existence, which by its own definition, cannot exist.

There is no 'own definition' in the first place; therefore there is no paradox.

If your reply is based on my first post of this thread then ignore it and instead
please read: http://www.geocities.com/complementarytheory/Russell1.pdf

Thank you,

Lama

 

[arildno]

Just a question, Lama:

You haven't by any chance read Hegel's "Wissenshaft der Logik"?
Several of your ideas seem to be in tune with Hegel's ideas..

 

[Lama]

Hi arildno,

No, I did not read any of Hegel's work.

Thank you for the information, I'll try to find an English version of it.

Can you give us some example, which shows the similarity between Hegel's work and my ideas?

Thank you.

Lama

 

[arildno]

Hi arildno,

No, I did not read any of Hegel's work.

Thank you for the information, I'll try to find an English version of it.

Can you give us some example, which shows the similarity between Hegel's work and my ideas?

Thank you.

Lama

Well, it's been years since I read Hegel, but he opposed, for example, to what he found was a philosophically incorrect concept of the limit.
(Basically, he meant the limit concept involved a "qualitative change" in the fundamental nature of the number)

Now, this is possibly of little interest/relevance to your own concepts, but I sensed a "resonance" of your ideas with Hegel's, rather than any specific concepts I could pinpoint at the instant.

 

[Lama]

Well my interpretation of the infinity concept is based on the complementary relations between the symmetry concept and the information concept.

Please read my answer to Gza: https://www.physicsforums.com/showpost.php?p=263204&postcount=69

and also my answer to Hyrkyl: https://www.physicsforums.com/showpost.php?p=263942&postcount=71


Also you can find my work in: http://www.geocities.com/complementarytheory/CATpage.html