Combinations

[Organic]

I realized that my last thread (named by "permutations") can’t be understood by professional mathematicians.

Because I don’t know how to write my idea in the common formal way, I am going to do it in a non-formal way, but I will do my best to write it in the clearest way.

So here it is:

Let us check these lists.

P(2) = {{},{0},{1},{0,1}} = 2^2 = 4

and also can be represented as:

00
01
10
11


P(3) = {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} = 2^3 = 8

and also can be represented as:

000
001
010
011
100
101
110
111

Let us call any full 01 list, combinations list.

Now, let us use Cantor's Diagonalization method on some finitely long combinations list, for example, the combinations list of number 3:

000
001
010
011
100
101
110
111

We can change the order of the rows, and then use Cantor's Diagonalization method, for example:

001
011
010
000
101
100
111
110

The input for Cantor's Diagonalization method in the first example is 000 and the output is 111.

The input for Cantor's Diagonalization method in the second example is 010 and the output is 101.

In both examples we find that the result is already in the combinations list, and this combination, which is already in the list, is one of the combinations that Cantor's Diagonal does not cover.

The number of the combinations, which are out of the range of Cantor's diagonal is:

2^n - n

Every column, which belongs to some combinations list is a sequence of 01 notations, based on some periodic frequency changes, for example:

the right column of number 3 combinations list, is based on 2^0(=1).

Therefore the periodic frequency changes are 1, and the result in this case is:
01010101.

The result of the middle column is based on 2^1(=2), therefore the sequence is:
00110011.

The result of the left column is based on 2^2(=4), therefore the sequence is:
00001111.

and we get the full combinations list of number 3:

000
001
010
011
100
101
110
111

We can get a combinations list of infinitely many places, by using the ZF Axiom of infinity induction, on the left side of our combinations list, by using the induction on the power_value of each column, for example:

2^0, 2^1, 2^2, 2^3, ...

In this stage we have proven, by induction, that Cantor's diagonal cannot cover any full 01 combinations list, finite or infinite.

Therefore its result is not a new combination (that has to be added to the list).

Because Cantor's diagonal cannot cover the full 01 combinations list (of aleph0 places for each combination) we can conclude that 2^aleph0 > aleph0.

But, because no diagonal's result is a new combination (and therefore not added to the list) each infinitely long sequence of 01 notations can be mapped with some natural number, for example:

...000 <--> 1
...001 <--> 2
...010 <--> 3
...011 <--> 4
...100 <--> 5
...101 <--> 6
...110 <--> 7
...111 <--> 8
...

Therefore we can conclude that 2^aleph0 = aleph0, and we come to contradiction.

(2^aleph0 >= aleph0) = {}, and we have a proof saying that Boolean Logic cannot deal with infinitely many objects in infinitely many magnitudes.

One can say that at least the sequence ...111 is not in the list, for example:

...000 <--> 1
...001 <--> 2
...010 <--> 3
...011 <--> 4
...100 <--> 5
...101 <--> 6
...110 <--> 7
...111 <--> 8
...

Let us examine the infinite from another point of view.

When we have ...111 AND ...000 in an ordered combinations list, it means that the list is complete.

But this is the whole point, infinitely many objects cannot be completed, otherwise they are finitely many objects.

Therefore ...111 AND ...000 are not in the list of infinity many objects.

In other words [...000, ...111) XOR (...000, ...111] .

There are 2 possible structural types of infinitely many 01 notations:

(?...0]
(?...1]

We know how some infinitely long combination starts, but its left side is
unknown (can be 0 XOR 1) and this missing information is essential to the existence of the induction.

Therefore we can find a meaningful missing result by Cantor's Diagonalization method, only in a finite combinations list.


For more details please look at:

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


Organic

[HallsofIvy]

The input for Cantor's Diagonalization method in the first example is 000 and the output is 111.

The input for Cantor's Diagonalization method in the second example is 010 and the output is 101.

As I said before, Cantor's Diagonalization method requires that you have as many digits in each number as you have numbers (the way it is normaly applied, this is a countable list of numbers each having a countable number of digits).

In addition, the "input" for Cantor's Diagonalization method is the list of numbers not any one number.

It is not a matter of not knowing how to "write" mathematics. You've never taken the time to understand the mathematics itself.

[Organic]

Dear HallsofIvy,


Now i know that you don't really want to understand what you read.

And to understand what i wrote you have to do 2 basic things:

1) You have to read it from the first word until the last word.

2) After you read all of it, you have to check if you understand it.

3) After you understand it, than and only than, please write a detailed reply.

Thank you.



Organic

[russ_watters]

Originally posted by Organic
Dear HallsofIvy,


Now i know that you don't really want to understand what you write.

And to understand what i wrote you have to do 2 basic things:

1) You have to read it from the first word until the last word.

2) After you read all of it, you have to check if you understand it.

3) After you understand it, than and only than, please reply.

Thank you.

Organic Frankly, Organic, the only thing that would keep me from laughing at the mess you made with that last thread would be my respect for you as a person (until proven otherwise, I assume everyone is worthy of my respect). Posts like this one diminish that. You won't get very far here unless you drop your attitude. And you will get even further if you first learn some real math before trying to invent your own new math.

[Organic]

Dear russ_watters,


First you have to show that you understand what i wrote in this thread, and you can do in this way:

1) You have to read it from the first word until its last word.

2) After you read all of it, you have to check if you understand it.

3) After you understand it, than and only than, please write your detailed reply.


If you can't follow these 3 steps, then you did not show to the persons that read your last raply, that you have any meaningful thing to say about what i wrote.


Yours,


Organic

[NateTG]

Perhaps you should try to understand Cantor's diagonal argument before you start claiming that it is invalid.

Usually the thing you call 'combinations list' in your post is referred to as 2^A.

Cantor's argument is that |2^A| \neq |A|. This should painfully clear if A is a finite set.

A particularly choice abuse is:
But each infinitely long sequence of 01 notations can be mapped with some natural number, for example:

...000 <--> 1
...001 <--> 2
...010 <--> 3
...011 <--> 4
...100 <--> 5
...101 <--> 6
...110 <--> 7
...111 <--> 8
...

Your example pairings are either not well defined, or for example, if the ...'s are all zeros will never map any sequence that contains an infinite number of 1's.

[suyver]

Originally posted by Organic
1) You have to read it from the first word until its last word.

I've read all of it.

Originally posted by Organic
2) After you read all of it, you have to check if you understand it.

I've understood what you are trying to say / prove. However, I do not agree with it.

Originally posted by Organic
3) After you understand it, than and only than, please write your detailed reply.

Your reasoning is flawed. Cantor's argument ONLY works for a list of n numbers, each containing exactly n digits. Since your list does not follow this requirement, you can't use Cantor.

[Guybrush Threepwood]

and I also didn't understand what do you mean by this......

Originally posted by Organic
In both examples we find that the result is already in the combinations list, and this combination, which is already in the list, is one of the combinations that Cantor's Diagonal does not cover.

The number of the combinations, which are out of the range of Cantor's diagonal is:

2^n - n
Organic [/B]

[Organic]

To NateTG,

Please show in what i wrote how do you come to the conclusion that what you call A, is finite.

[Organic]

To suyver,

You are right, Cantor's diagonal is limited to aleph0^2(=aleph0), and this is exactly the reason why he can find infinitely many new numbers, which are not in this aleph0^2(=aleph0) list.

The complete list of all R numbers is 2^aleph0.

I proved, by using the ZF axiom of infinity induction, that 2^aleph0 > aleph0.

Then i proved that 2^aleph0 = aleph0 by this:

Each infinitely long sequence of 01 notations can be mapped with some natural number, for example:

...000 <--> 1
...001 <--> 2
...010 <--> 3
...011 <--> 4
...100 <--> 5
...101 <--> 6
...110 <--> 7
...111 <--> 8
...

Therefore we can conclude that 2^aleph0 = aleph0, and we come to contradiction.

(2^aleph0 >= aleph0) = {}, and we have a proof saying that Boolean Logic cannot deal with infinitely many objects.

[Guybrush Threepwood]

Originally posted by Organic
I proved, by using the ZF axiom of infinity induction, that 2^aleph0 > aleph0.


how???

http://www.mtnmath.com/book/node53.html

[Organic]

To Guybrush Threepwood,

You wrote:

and I also didn't understand what do you mean by this......


Here it is again:

Now, let us use Cantor's Diagonalization method on some finitely long combinations list, for example, the combinations list of number 3:

000
001
010
011
100
101
110
111

We can change the order of the rows, and then use Cantor's Diagonalization method, for example:

001
011
010
000
101
100
111
110

The input for Cantor's Diagonalization method in the first example is 000 and the output is 111.

The input for Cantor's Diagonalization method in the second example is 010 and the output is 101.

In both examples we find that the result is already in the combinations list, and this combination, which is already in the list, is one of the combinations that Cantor's Diagonal does not cover.

The number of the combinations, which are out of the range of Cantor's diagonal is:

2^n - n

[Guybrush Threepwood]

Organic, you don't have to repaste the original message.[6)]


In both examples we find that the result is already in the combinations list, and this combination, which is already in the list, is one of the combinations that Cantor's Diagonal does not cover.


If the combination is already on the list, what do you mean is not covered?

And also the argument HallsOfIvy made about the incorrect use of the diagonalization method in this case stands.

[Organic]

I proved it for n, i proved it for n+1, therefore i proved it for infinitely many cases.

[Organic]

Please read my answer to To suyver.

[HallsofIvy]

I proved it for n, i proved it for n+1, therefore i proved it for infinitely many cases
Now, that's just bad elementary mathematics!

[Guybrush Threepwood]

Originally posted by Organic
To suyver,

I proved, by using the ZF axiom of infinity induction, that 2^aleph0 > aleph0.


I ask again how?? I provided a link to the ZF set of axioms earlier. I just don't see it....


Each infinitely long sequence of 01 notations can be mapped with some natural number, for example:

...000 <--> 1
...001 <--> 2
...010 <--> 3
...011 <--> 4
...100 <--> 5
...101 <--> 6
...110 <--> 7
...111 <--> 8
...


no, you forgot at least the all '1' sequence.....

[Organic]

Dear HallsofIvy,

Ok, show in detailed reply why it is bad elementary mathematics.

[Organic]

Let us examine the infinite from another point of view.

When we have ...11111 AND ...00000 in an ordered combinations list, it means that the list is complete.

But this is the whole point, infinitely many objects cannot be completed, otherwise they are finite.

Therefore ...11111 AND ...00000 are not in the list of infinity many objects.

In other words [...000, ...111) XOR (...000, ...111] .

There are 2 possible structural types of infinitely many 01 notations:

(?...0]
(?...1]

We know how some infinitely long combination starts, but its opposite side is
unknown (can be 0 XOR 1) and this missing information is essential to the existence of the induction.

Therefore we can find meaningful missing result by Cantor's diagonal method, only in a finite combinations list.

For more details please look at:

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

[HallsofIvy]

Dear HallsofIvy,

Ok, show in detailed reply why it is bad elementary mathematics.
The statement "I proved it for n, i proved it for n+1, therefore i proved it for infinitely many cases." doesn't make any sense.
If you mean "n" as a single specific integer, then "I proved it for n, i proved it for n+1" only proves it for those two particular numbers.
If you meant n to be any natural number, then after "I proved it for n" it isn't necessary to prove it for n+1!
If you meant this as "proof by induction", then it is "Assuming the statement is true for n, prove it is true for n+1" and is only part of what is necessary for a proof by induction.

[Organic]

Hi HallsofIvy,

Please go to the frist post of this thead, find the part of my proof by induction, and then please tell me if this part is proof by induction or not, and if not, why not.

Thank you,

Organic

[NateTG]

Might be a poorf, definitley not a proof. [;)]

That one misses sequences like:
...01010101010101
which contain infinitely many ones and zeros.

[Organic]

Thank you NateTG, I deleted it.

Please show in what i wrote how do you come to the conclusion that what you call A (look in page 1), is finite.




Organic

[HallsofIvy]

Okay, I did go back and read it. You look at 2 bit sequences and show that the number you get by applying Cantor's method to the sequence is in the sequence. Then you look at 3 bit sequences and show that the number you get by applying Cantor's method to the sequence is in the sequence. You then assert that you have proved "by induction" that this will be true for any finite or infinite.
The simplest thing that is wrong with that is that you did not show that "if it is true for N then it is true for N+1".

Actually my objection was to your statement "I proved it for n, i proved it for n+1, therefore i proved it for infinitely many cases."
Even assuming you meant "I proved that IF it is true for any positive integer n, then it is true for n+1, therefore I proved it true for infinitely many cases." that would not be correct- you still have to prove it is true for onecase.(And by the way, induction does not just prove "it is true for infinitely many cases", it proves the statement (which depends on the positive integer n) is true for all n. Saying something is true for all positive integers n is far different than saying it is true for infinitely many of them.)

Somewhat less simple is this: induction proves something is true for all positive integers- not for infinity. You referred to "ZF axiom of infinity induction" but you did not use that. A proof by infinite induction would have to appeal to Zorn's lemma or something equivalent.

Finally, you basic concept of Cantor's method is flawed. There are 4 binary numbers with 2 bits. In applying (your version of) Cantor's method, you used only 2 of them. Of course, the number you got was one of the other 2. There are 8 binary numbers with 3 bits. In applying Cantor's method, you used only 3 of them. Of course, the number you got was one of the other 5. In any list of numbers with n bits, there are 2n numbers, of which you would use the first n to produce a "new" number. It should be no surprise that that number is in the other 2n- n.

Cantor applied his method to a list of real numbers, each of which was represented by an infinite (countable) number of digits (or bits if you want to use binary notation). In constructing his new number, with an infinite number of bits, he used all of the numbers in the list. Since the nature of the construction guarenteed that the new number could not be any of those used in its construction, and all numbers in the list were so used, it follows that the new number cannot be on the list. That is the argument that does not apply to any finite list of numbers.

[Organic]

Dear HallsofIvy,

Thank you for your detailed reply.

The ZF Axiom of infinity simply says: if n exists than n+1 exists.

I use this built-in induction on the power_value of 2^power_value,
and the result of using the built-in induction of the ZF Axiom of infinity on the power_value, cannot be but 2^aleph0.

Please show me some mistake in what i wrote in this post.

Yours,


Organic

[NateTG]

Originally posted by Organic
Thank you NateTG, I deleted it.

Please show in what i wrote how do you come to the conclusion that what you call A (look in page 1), is finite.

Organic

Originally posted by NateTG
Cantor's argument is that |2^A| \neq A. This should be painfully clear if A is finite

In that phrase "A is finite" is intended to be a hypothesis, not a conclusion.

Please do not wipe out posts that people have responded to. It makes the thread difficult to follow.

Because Cantor's diagonal argument is constructive, it's easy to give an example sequence of 1's and 0's that any particular function f:\mathbb{N}\rightarrow \{0,1\}^{\mathbb{N}} does not cover.

[NateTG]

Originally posted by Organic
Dear HallsofIvy,

Thank you for your detailed reply.

The ZF Axiom of infinity simply says: if n exists than n+1 exists.

I use this built-in induction on the power_value of 2^power_value,
and the result of using the built-in induction of the ZF Axiom of infinity on the power_value, cannot be but 2^aleph0.

Please show me some mistake in what i wrote in this post.

Is your intent is to claim something resembling
\lim_{n\rightarrow \aleph_0} 2^n = \aleph_0?

[Organic]

Hi NateTG,


Please do not wipe out posts that people have responded to. It makes the thread difficult to follow.

You are right, i am sorry.

By using the built-in induction of the ZF axiom of infinity, on the power_value of base 2, we get two things:

1) Because the axiom says "if n exists then n+1 exists", the rusult of using it cannot be but 2^aleph0.

2) We get an odered list of (2^aleph0)-1 unique combinations of 01 notations, where the diagonal can cover only aleph0 notations.

Therefore we cannot conclude that the diagonal result is a new sequence (that has to be added to the list).

But the important insight is this:

When we have ...111 AND ...000 in an ordered combinations list, it means that the list is complete.

But this is the whole point, infinitely many objects cannot be completed, otherwise they are finitely many objects.

Therefore ...111 AND ...000 are not in the list of infinity many objects.

In other words [...000, ...111) XOR (...000, ...111] .

There are 2 possible structural types of infinitely many 01 notations:

(?...0]
(?...1]

We know how some infinitely long combination starts, but its opposite side is unknown (can be 0 XOR 1) and this missing information is essential to the existence of the induction.

(
By the way, Rational-like infinitely many sequences, has 4 possible structural types:

0...0
0...1
1...0
1...1
)


Therefore we can find a meaningful missing result by Cantor's Diagonalization method, only in a finite combinations list.


For more details please look at:

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


Organic

[moshek]

we must read the way Hilbert end his lecture at paris 1900
to see how much this topic in So significant.

[Organic]

Dear Moshek,

Thank you !!

the way Hilbert ended his lecture at paris 1900, can be found here:

http://babbage.clarku.edu/~djoyce/hilbert/problems.html


No organic system is a closed system.


The "key word" for any open information's system is 'uncertainty'.

'Completeness' and 'infinitley many objects' are complementary concepts (like waves and particles in Quantum Mechanics).

Therefore, no infinitely many objects can be completed and well known like finitely many objects.

Please read this again:

By using the built-in induction of the ZF axiom of infinity, on the power_value of base 2, we get two things:

1) Because the axiom says "if n exists then n+1 exists", the rusult of using it cannot be but 2^aleph0.

2) We get an odered list of (2^aleph0)-1 unique combinations of 01 notations, where the diagonal can cover only aleph0 notations.

Therefore we cannot conclude that the diagonal result is a new sequence (that has to be added to the list).

And because nothing is added to the list, we can conclude that (2^aleph0)-1 = aleph0.

And also we can conclude that (2^aleph0)-1 > aleph0, because the diagonal can cover at most aleph0 notations.

Therefore ((2^aleph0-1) >= aleph0) = {}.

Again:

When we have ...111 AND ...000 in an ordered combinations list, it means that the list is complete.

But this is the whole point, infinitely many objects cannot be completed, otherwise they are finitely many objects.

Therefore ...111 AND ...000 are not in the list of infinity many objects.

In other words [...000, ...111) XOR (...000, ...111] .

There are 2 possible structural types of infinitely many 01 notations:

(?...0]
(?...1]

We know how some infinitely long combination starts, but its opposite side is unknown (can be 0 XOR 1) and this missing information is essential to the existence of the induction.

(
By the way, Rational-like infinitely many sequences, has 4 possible structural types:

0...0
0...1
1...0
1...1
)


Therefore we can find a meaningful missing result by Cantor's Diagonalization method, only in a finite combinations list.

[Hurkyl]

(For the record, just because I'm not responding to something doesn't mean it's not wrong nor a rampant abuse of language)

There are 2 possible structural types of infinitely many 01 notations:

(?...0]
(?...1]

We know how some infinitely long combination starts, but its opposite side is unknown (can be 0 XOR 1)

Correction: there is no "opposite side". An infinite sequence has at most one endpoint.

[Organic]

Hi dear Hurkyl,


For me you are the best mathematician in this forum.


Please read my first post, and write your detailed remarks on each part of it.


Thank you.

Yours,


Organic


Correction: there is no "opposite side". An infinite sequence has at most one endpoint.

Rational-like sequences:

(...010010] = (0...0]
(...010101] = (0...1]
(...101010] = (1...0]
(...101101] = (1...1]

[moshek]

Dear Organic.

Thank you very much for the direction to the way Hilbert end his lecture at Paris:


"The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfill this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples"

(D.Hilbert 1900)



I am Sorry but I don’t think yet that there is a mistake in Cantor diagonal meted. But I want ot ask you:

Will you be really satisfy if you convict us that there is some problem in Cantor argument.

thank you
Moshek


[:)] [:)]

[Organic]

Hi moshek,

I do not convict anyone in anything, all what i want is to share my ideas with other persons.

Please let me show you some interesting connection between redundancy and uncertatinty, when we construct the combinations list, by using the ZF axiom of infinity.

For example, let us look at 2^2:

0 0
0 1
----
1 0
1 1

And now let us look at 2^3:

0 00
0 01
0 10
0 11
-----
1 00
1 01
1 10
1 11

And 2^4:

0 000
0 001
0 010
0 011
0 100
0 101
0 110
0 111
------
1 000
1 001
1 010
1 011
1 100
1 101
1 110
1 111


In all examples the uniquness of each row, depends on the left most 0 XOR 1 notations.

But when we have infinitely many 01 notations in each row, the left most 0 XOR 1 notations cannot be reached by us, thefore it is unknown, and we always have two identical lists, that cannot be distinguished from each other.

Both uncertainty and redundancy values depends on the number of different notations in any combinations list, for example:

2={'0','1'} , 3={'0','1','2'} , 4={'0','1','2','3'} , ...


I think because of this connection between uncertainty an redundancy (when dealing with infinity), Cantor's Diagonalization method cannot work on infinitely many objects.

This is the main idea of my proof.

Uncertainty and redundancy are essential properties of any rigorous argument dealing with infinitely many objects.

'Completeness' and 'Infinitely many objects' are complementary concepts (exactly like waves and particles in Quantum Mechanics).

When you don't internalize it, then there is no connection between our point of views, about the infinity.




Organic

[moshek]

Organic,

I am really glade that you don’t try to convict me that there is new way to look on mathematics, even if you think so. I will think also and replay to you in about two days.

Thank you
Moshek
[zz)]

[Organic]

Again, we use the built-in induction of the ZF axiom on the power_level of 2^0, 2^1, 2^2, ...

Because of the uncertainty and redundancy poroperties, we cannot talk about ALL obejcts in a collection of infinitely many objects.

The most we can say is: power_value approaches(-->) aleph0.

Again:

Uncertainty and redundancy are essential properties of any rigorous argument dealing with infinitely many objects.

'Completeness'(ALL objects of some collection) and 'Infinitely many objects' are complementary concepts (exactly like waves and particles in Quantum Mechanics).

Therefore to say that |N|=aleph0 is as if we say:

1(='completeness') XOR 1(='infinitely many objects') is 1.

[moshek]

Sorry Organic there is no problem in Cantor prove that |P(N)|>|N|
and what you say is your prove, is not a regular mathematics prove at all.

But Still maybe you see something important!

I want to ask you why do you think it is worth today after Hibert recognize in 1900 with the cardinals in his first probelm (The 23 list) to look again on Cantor work is the way you want to direct it?

[Organic]

Hi Moshek,


I am not talking about some thechnical problem, but on the essential property that distinguishes between what we call potential and actual infinity.

In my opinion Cantor did not distinguished between them when he developed its mathematical system.

In my opinion, any Math system is first of all an information system.

If no input then no output and no any meaningful conclusion.

In the case of the cardinality of N, |N| approaches(-->) aleph0.

When |N|=aleph0 we have no information bacause no infinitely many objects can reach their limit.

|N|approaches(-->)aleph0 is what we call a potential infinity.

|N|=aleph0 is what we call an actual infinity.

In an actual infinity you cannot find any information of any kind.

Therefore Math language, which is first of all an information system, can deal only with a potential infinity.

Please look at this again: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Organic

[Hurkyl]

|N| is a fixed constant. How does it approach something?

[moshek]

That is a very good question to Organic.

So let's wait to his answer![6)]

[Organic]

by ZF axiom of infinity all we can say is:

Omega={0,1,2,3,4,5,6,7,....}

because of the ,....} notations we cannot conclode that Oomega=Actual infinity.

More than thet, When Omega=Actual infinity then:

Omega=


There is no meaningful information when we force the word 'ALL'(=complete) on 'infinitely many objects'(=cannot be completed).

Basically we can distinguish between 3 states:

1) All, complete for finite information.

2) Infinitely many objects for potential infinity.

3) No information for actual infinity

Therefore, if we want that |N|=Actual infinity, then |N|=

Organic

[Guybrush Threepwood]

Originally posted by Organic
by ZF axiom of infinity all we can say is:

Omega={0,1,2,3,4,5,6,7,....}

because of the ,....} notations we cannot conclode that Oomega=Actual infinity.


are you trying to say that \mathbb{N} doesn't have \aleph_0 elements? that \mathbb{N} is potentially infinite, but actually finite?

[Organic]

When by writing aleph0 we mean that aleph0=actuacl infinity, if we want |N| to have a meaningful information, then |N| approaches(-->) aleph0(=actual infinity).

Again, when |N|=aleph0=actual infinity, then |N|=(no information of any kind)

Please look at this example:

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



Organic

[moshek]

I am realy afraid that Organic
want to tell us such a thing.

Does Organic
is realy a monkey
like he is looking ?[o)]

[Guybrush Threepwood]

just because we can't actually count to infinity doesn't mean infinity doesn't exist...........

moshek: that's not very nice of you

[Organic]

Hi Moshek,

When i am looking on your picture, i know that Moshek=Actual infinity.



Orgainc

[moshek]

I hope i didn't heart you
i realy like the way you think
So i just want to triger you.

well i dont have any picture
like that you have,
but i want to tend to infinity
even more than Alef0

So realy thank you Organic.
[:))]

[Organic]

Dear Guybrush Threepwood,

Let us think about these 4 possible contents:


{} = Emptiness.

{1,2} = Finite or complete content.

{1,2,...} = Infinitely many objects(=cannot be completed).

{______} = Fullness = Actual infinity(=cannot be factorized to any form of information).




Orgainc

[Organic]

Hi Moshek,


Dont take it to your heart, i like your picture (i mean actual infinity).


Yours,


Organic

[moshek]

Dear Organic,

Set theory is a very beautiful theory in mathematics
about the infinity !

Close to the end of the 19 century
the lord Kelvin pointed to 2 problem in physics
that are unsolved and they were solved in 1905..

Which problems in mathematics
you are trying to solved?

[:)]

[Organic]

There are two basic forms of set's contents where the word 'many' is meaningless:

{} Eemptiness

{_} Fullness

These two basic forms Cannot be reached by finite or infinitely many objects.

Finite or infinitely many ojbects can only approach these two basic forms of set's contents.

Therefore 0 and oo (or -oo) are the limits( (0,oo) or (-oo,0) ) of any information system, including Math language.

And again, it is clearly shown here:

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

When these two unreachable and opposite limits are associated, new forms of information can be defined, explored by us, and used to help us be better participators in this universe.

Therefore concepts like complexity, uncertainty and redundancy, based on simple principles, have to be taken as natural basics of any axiomatic system.

By doing that, i think Math language can be developed to variety of unexpected areas, beyond our wildest dreams.

[Hurkyl]

|\mathbb{N}| is something. *shrug*

[phoenixthoth]

By doing that, i think Math language can be developed to variety of unexpected areas, beyond our wildest dreams.

i agree. i wonder where math will be a millenium from now...

[Organic]

|N| is the cardinal of infinitely many objects that approaching aleph0, where aleph0=actual infinity.

[phoenixthoth]

actually, |N|=aleph0 and nothing is approaching anything.

but if we let f be a map from N to P(N) such that f(n) is the set of elements in N less than n+1, then f(0), f(1), f(2), ... in some sense approaches N.

|N|=aleph0.

[moshek]

Hurky - I did not understood the word that you wrote after |N|
can you explain it to me?

Phoenixthoth- To really discus how the new millennium will be for mathematics we must go back to Euclid and ask him some question!
About his "Elements".

Maybe this is what Organic is trying to do here.

Moshek [;)]

[phoenixthoth]

can you explain it to me?
sounds like a question clinton would ask. not as simple as one might think, actually.

[moshek]

Sory Phoenixthoth [:((]
Moshek

[Organic]

Hi phoenixthoth,

By using the word 'approaching' i don't mean 'closer to'.

'approaching' = 'closer to' only on a finite collection of objects.

When i use 'approaching' with infinitely many objects, then
'approaching aleph0' = 'cannot reach aleph0'.

Therefore (|N|=aleph0) = (|N|={____}=fullness) = No meaningful information's input.

Math language cannot deal with {}(=emptiness) XOR {___}(=fullness) contents.


Question 1: How many times we can reach 2 in {1,2}?

Answer 1: Infinitely many times.

Questions 2: How many time we can reach aleph0 by using {1,2,3,...}?

Answer 2: 0 times.

[HallsofIvy]

The problem with saying that "Math language cannot deal with {}(=emptiness) XOR {___}(=fullness) contents." is that you clearly know neither mathematics or "math language" and so have no business talking about what math language can or cannot deal with.

I will, however, concede that math language cannot, in fact, deal with nonsense.

[Organic]

Hi HallsofIvy,



Please show us how you can use the content of {}(=emptiness) or the content of {__}(=fullness) as an input, by Math language.

[Hurkyl]

Organic: are you simply trying to say \aleph_0 is not a natural number?


Please show us how you can use the content of {}(=emptiness) or the content of {__}(=fullness) as an input, by Math language.

Like HallsofIvy said, math can't deal with nonsense.

[Guybrush Threepwood]

Originally posted by Organic
Please show us how you can use the content of {}(=emptiness) or the content of {__}(=fullness) as an input, by Math language.


|{}| = 0

[Organic]

Hi Hurkyl,

Any information system needs some input, and Math is a form of information system.

There are at least two concepts which are the limits of any information system, including math.

(emptiness,fullness) no input can be found beyond these limits.

Therefore Cantor's idea about aleph0 is nonsense, because he does not distinguish between actual and potential infinity.

[Organic]

Guybrush Threepwood,

Without the set notations '{' '}' you cannot do that.

I am talking about the emptiness(the content) itself and the fullness(the content) itself.


For example:

=?
_=?

[master_coda]

Originally posted by Organic
Any information system needs some input, and Math is a form of information system.

There are at least two concepts which are the limits of any information system, including math.

(emptiness,fullness) no input can be found beyond these limits.

Therefore Cantor's idea about aleph0 is nonsense, because he doeas not distinguish between actual and potential infinity.

Just saying these things doesn't make it so.

You haven't provided a rigorous definition of anything. And you haven't demonstrated that you have any understanding of the mathematical terms that you use.

[Organic]

master_coda,

Please look at:

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

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

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

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

Also please read ALL this thread and then reply.


Organic

[Guybrush Threepwood]

Originally posted by Organic
Guybrush Threepwood,

Without the set notations '{' '}' you cannot do that.

I am talking about the emptiness(the content) itself and the fullness(the content) itself.


I really don't understand your point. Mathematics works with symbols. {} is one of them. So is \aleph_0 or \mathbb{N}
If you want to define a new symbol please do so and say what it means. If you want to do mathematics without symbols I'm afraid that's not possible

[Organic]

A symbol is a tool, if you dont understend the meaning of the concept that you notate, then you can invent and use any notation that you want, but the meaning of it is beyond the notation.

Again i clime that when Cantor invented the aleph0 notation, he did it without distinguishing (sorry about my English) between actual and potential infinity.

I'll write this again:

by ZF axiom of infinity all we can say is:

Omega={0,1,2,3,4,5,6,7,....}

because of the ,....} notations we cannot conclode that Omega=Actual infinity.

More than thet, When Omega=Actual infinity then:

Omega=


There is no meaningful information when we force the word 'ALL'(=complete) on 'infinitely many objects'(=cannot be completed).

Basically we can distinguish between 4 states:

0) Emptiness (no information).

1) All, complete, for finite information.

2) Infinitely many objects for potential infinity.

3) No information for actual infinity

Therefore, if we want that |N|=Actual infinity, then |N|=
and the same is about aleph0.


Organic

[master_coda]

Originally posted by Organic
Again i clime that when Cantor invented the aleph0 notation, he did it without distinguishing (sorry about my English) between actual and potential infinity.

Distinguishing between "potential infinity" and "actual infinity" implies that infinity is some sort of process. It is not. Terms like \aleph_0 have a very well defined meaning already, one that does not need to be furthur distingished.

[Organic]

master_coda,


Please read this: http://www.geocities.com/complementarytheory/SPI.pdf

As we can see from the example, we are in a "never ending story"
of extrapolation, which is limited by {__}, and interpolation, which is limited by {}.



Terms like aleph0 have a very well defined meaning already, one that does not need to be furthur distingished.


By writing these words you say that Math language is a closed information system.

Do you know what is the destiny of closed systems?

[Tempest]

I don't have the time right now to fully understand the original post organic, and I'm not really sure what you are trying to prove. Are you trying to prove that cantor made an error in his proof that there are more numbers in the interval [0,1] than natural numbers?

Let your numeral system be binary with 0<1

Consider all possible representations of natural numbers

0
01
10
11
100
101
110
111
1000
1001

etc

Now, consider all possible sequences of zeros and ones next to the 'decimal' point. Really it needs to be called a 'binal' point or something.

0.1000000000...
0.0100000000...
0.1100000000...
0.0010000000
0.0110000000
etc.

My point would be, there are as many sequences to the right of the 'binal' point as there are to the left of the 'binal' point. And even Pi has some binal representation of zeros and ones, and so since there is a one to one mapping of sequences from the right side of the binal point to the left side, there are just as many numbers in [0,1] as there are natural numbers. I mean I know this was sloppy, but whatever.

[Organic]

hi Tempest,

When you have the time please read again the first post, and if you have more time then please read the whole thread, and then please reply.


Thenk you.


Organic

[Tempest]

But its sooooo complicated Organic. Can you at least tell me what you are trying to prove?

[Organic]

Ok, I'll try.

One of the axioms of Axiomatic set theory is the Axiom of infinity, which gives us the ability to deal with infinitely many objects.

This axiom simply says: if n exists then n+1 exists.

This axiom is based on a built-in induction, and i use this property to show that there is a fundamental problem in the infinity concept, as it is used by Modern Math, which is based on Cantor's mathematical approach.

In my opinion, Cantor did not distinguish between two different types of infinity, which are: Actual infinity and Potential infinity.

If we look on Math language as a form of information system, then Actual infinity is the limit of any information system, including Math language.

For example, the most simple object in set theory is the empty set, which means, a set with no content that notated as {}.

"Below" emptiness there is no information, therefore emptiness is the lowest limit of Math language.

Is there an highest limit to Math language ?

When i researched this question i have found that by using the actual infinity concept, we define that there is an opposite concept to emptiness, which is fullness that can be notated as {__}, where "above" it there is no information.

Therefore fullness is the highest limit of Math language.

Also i have found that Cantor used words like 'all' and 'complete'
in a wrong way, by connecting them to the infinity concept.



Explanation 1:

Let us think about these 4 possible contents:

{} = Emptiness.

{1,2} = Finite or complete content.

{1,2,...} = Infinitely many objects(=cannot be completed).

{______} = Fullness = Actual infinity(=cannot be factorized to any form of information).



Explanation 2:

by ZF axiom of infinity all we can say is:

Omega={0,1,2,3,4,5,6,7,....}

because of the ,....} notations we cannot conclude that Omega=Actual infinity.

More than that, When Omega=Actual infinity then:

Omega=no information


There is no meaningful information when we force the word 'ALL'(=complete) on 'infinitely many objects'(=cannot be completed).

Basically we can distinguish between 4 states:

0) Emptiness (no information).

1) All, complete, for finite information.

2) Infinitely many objects for potential infinity (cannot be completed).

3) No information for actual infinity.

Therefore, if we want that |N|=Actual infinity,
then |N|=no information, and the same is about aleph0.

To make it clearer, please look at:

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

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

and also:

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

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

and again:

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


Question 1: How many times we can reach 2 in {1,2}?

Answer 1: Infinitely many times.

Question 2: How many time we can reach aleph0 by using {1,2,3,...}?

Answer 2: 0 times.



Yours,


Organic

[master_coda]

Originally posted by Organic
Question 2: How many time we can reach aleph0 by using {1,2,3,...}?

Answer 2: 0 times.

We are not trying to "reach" aleph0. Aleph0 is defined as the cardinality of the natural numbers.

I've already said this: infinity has a very well defined meaning in math, one that you do not seem to understand. Whatever definition of infinity you are using, it has nothing to do with the mathematical definition.

[Organic]

Hi master_coda,

Please show us the well defind meaning of aleph0.

Maybe this can help:

http://en2.wikipedia.org/wiki/Absolute_Infinite

http://mathworld.wolfram.com/Aleph-0.html

http://mathworld.wolfram.com/TransfiniteNumber.html

http://mathworld.wolfram.com/CardinalNumber.html

[Hurkyl]

One of the things that makes mathematics such a powerful tool is that it requires concepts to be expressed in a precise way.

When you write your idea in a precise form, it allows you (and others) to see what your idea "says", and possibly to find that its flawed. If you don't like what it says or find that its flawed, you discover something wrong with your mental formulation of your idea and you change the precise form to try to come up with something better.

The other great thing about having your idea in a precise form is that, even if someone else thinks your concept is nonsense, 'e can't argue with the math, if you've done it correctly.


And that's the main point here; as far as everyone knows, set theory has been done correctly. You may not like set theoretic concepts like cardinal numbers, but they have a precise definition and they work. (Of course, you're free to intuit a cardinal number as whatever you like)


The other main point is that you are not giving any attempt at expressing your idea in a precise way; you seem to be interested in only giving unjustified, nebulous statements then using them to "disprove" standard results.

If you want to spend the effort to develop a new set theory, fine. If you want to spend the effort to develop your ideas in terms of existing set theory fine. What you have been doing for the past months is not fine.

[Organic]

Hi Hurkyl,

In the last 4 months i was hopping to get some help from professional mathematicians in this forum.

You helped me in this idea: http://www.geocities.com/complementarytheory/ET.pdf
and i really want to thank you for this.

I also wrote this: http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf

As a response i got a lot of personal replies (telling me how i am and where i should go), and all this time i was still hopping that some person will take an idea of mine and together through positive attitude, we shall try to check if the idea can survive a rigorous formal definitions, or not.

I discovered that the ability of most mathematicians to understand and translate ideas out of the common Math vocabulary, to common mathematical vocabulary, is very low.

Most of them choose the easy way, which is: If you don’t know the common mathematical vocabulary, then you cannot think Math.

Let us speak on this thread.

Instead of telling me in a general way if what i am doing is fine or not, please change your attitude, be more specific, use your professional skills and show me in a detailed way why my ideas are not fine:

Here are my ideas from the last 4 months:

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

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

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

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

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

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

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


Please use your skills to give a constructive criticism on the above ideas.


Thank you very much for your help.



Organic

[Guybrush Threepwood]

Originally posted by Organic
I discovered that the ability of most mathematicians to understand and translate ideas out of the common Math vocabulary, to common mathematical vocabulary, is very low.


well, nothing personal, but you don't express your ideeas in common Math vocabulary.
I mean what about actual infinity and potential infinity.... how do you define them, how are they different? In mathematics you have to define the terms you are using.....

PS: an please put some htmls on your site instead of pdfs....[:)]

[Organic]

Hi Guybrush Threepwood,

How do you come to the conclusion that i did not define actual and potential infinity?

Here it is written in a very clear (but a non-formal) way:

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

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

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

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


Please use you skills as a mathematician to give a constructive criticism on the above ideas.

(PDF format is the best way today to represent papers)

Yours,


Organic

[Guybrush Threepwood]

Organic, believe me I read all you pdfs.
In none of them I found something like: "actual infinity is ....." and I really don't understand what are you refering to by those names. Let me ask you something:

is the set {1, 2, 3} finite or not?
is the set {1, 2, 3, ... , \infty} = \mathbb{N}^* finite or not?
is the set {0, 2, 4, 6, ... , \infty} finite or not?
is the set {-\infty, ..., -3, -2, -1, 0, 1, 2, 3, ... , \infty} = \mathbb{Z} finite or not?

if any of the above is infinite please say what type on infinite (potential or actual) and why?

[Hurkyl]

{1, 2, 3, ... , \infty} = \mathbb{N}^*
is the set {-\infty, ..., -3, -2, -1, 0, 1, 2, 3, ... , \infty} = \mathbb{Z}

These equalities are incorrect; neither \mathbb{N} nor of \mathbb{Z} have an element named \infty.



As a response i got a lot of personal replies (telling me how i am and where i should go), and all this time i was still hopping that some person will take an idea of mine and together through positive attitude, we shall try to check if the idea can survive a rigorous formal definitions, or not.

As I recall, most progress was made when we started talking about your ideas and stopped talking about why your ideas "disprove" mathematics. That is the point I'm trying to make here.

[Organic]

Dear Hurkyl,


...What you have been doing for the past months is not fine.


I dont now what to say, it is too general.

For example, please read this:

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

and please give your detailed and constructive criticism on it,
as you started to do in:

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


This is all what i hope, to communicate with you on this subjects.

No more, no less.


Yours,


Organic

[Organic]

Hi Guybrush Threepwood,

First, please read Hurkyl's reply.

Actual infinity is the opposite of Emptiness {}, which means Fullness {___}.

These contents are the limits of any information system, including Math language.




Organic

[master_coda]

But saying "Fullness {___}" is not anymore meaningful than saying "actual infinity". You're trying to define terms using other undefined terms. And "opposite" isn't a mathematical term either, so you can't use it to define new terms.


There are a lot of situations in math where you can take a vague, more intuitive idea, and then try and provide a rigorous, mathematical definition. For example, continuity and connectedness. The process goes something like this:

1. Take an intuitive idea.
2. Find a mathematical way to express that idea.
3. See what conclusions you can draw from your new mathematical definition.

But you're skipping step two. You're applying your intuitive ideas to traditional math, without first expressing your ideas rigorously. You can't start drawing logical conclusions without first rigorously defining your starting points.

[Guybrush Threepwood]

Originally posted by Hurkyl
These equalities are incorrect; neither \mathbb{N} nor of \mathbb{Z} have an element named \infty.


sorry, next time I'll just let a lot of .................... and Organic will begin to say againg that is aproaching \aleph_0 and I will still not understand what he's talking about [t)] [t)]

Organic I'm still waiting for your response on which of the sets are actual and which potential infinities.....

[Hurkyl]

Is there any x such that x \in \{\_\_\}?

[Organic]

Dear Hurkyl and Guybrush Threepwood,

Please look at this:

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


Thank you.



Organic

[Guybrush Threepwood]

[*(] [*(] [*(] [*(]
no.....more....pdfs....please....

[master_coda]

Providing pdfs with even more undefined terms makes it harder to understand what you are talking about.

[Organic]

Hi master_coda,

Please give me an example of an undefined term in this pdf:

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

and then we shall try together (if you want) to address it in a formal way.

Thank you.


Organic

[NateTG]

Here's a short list:

limits
potential infinity
actual infinity
emptiness
fullness
limited information model
symetric potential infinity
directed potential infinty
floating point
floating point system
extrapolation
interpolation
scales
information cells
notated information cells
specific direction
aleph0
{}
{__}

I assume that integers part and fractions part, you mean integer part and fractional part, otherwise those are also not well-defined.

Some of the terms have common mathematical definitions, but you do not use them in ways that make sense using those definitions. For example "- aleph0" does not make any sense if you mean the cardinal number \aleph_0

[moshek]

It is known that among the list
of the 23 problem of Hilbert ( 1900 Paris)
only 3 left unsolved : 6,8,16.

So I just want to ask Organic:

Does your study in matematics
is relate to any of them?

thank you
Moshek[t)]

[Organic]

Hi NateTG,

Thank you for your reply.

Please take this by the standard mathematical meaning:

limits
emptiness
floating point
{}
extrapolation
interpolation
scales

The other concepts can be understood from the examples in the last pdf file.

Please fill free to ask any question that you like about them.

Thank you,


Organic

[Organic]

Hi moshek,

My ideas are deeply connected to the 6th problem.

[moshek]

Thank you for sharing that.

Moshek[:)]

[master_coda]

Originally posted by Organic
Hi NateTG,

Thank you for your reply.

Please take this by the standard mathematical meaning:

limits
emptiness
floating point
{}
extrapolation
interpolation
scales

The other concepts can be understood from the examples in the last pdf file.

Please fill free to ask any question that you like about them.

Thank you,


Organic

Emptiness does not have a mathematical meaning.

[moshek]

On which mathematics emptiness have no meaning?

[Organic]

Hi master_coda,

Emptiness is the content of {}.

Without it Axiomatic set theory can't hold.

[moshek]

Hi Organic,

I did not understand until now,
That you wan't axiomatic set theory to hold as it is today?

Are you please with P.Choen forcing method
to solve Hilbert first problem CH ?

please explain that to me.

Moshek
[t)]

[master_coda]

{} is the empty set. Emptiness is a word that has a great deal of philosophical baggage that adds confusion to the issue.

For example, you seem content to define fullness as the opposite of emptiness. After all, fullness is clearly the opposite of emptiness.

Except that it isn't rigorous at all. What is fullness? The set of all sets? The set of all things?

[Organic]

Hi moshek,

CH problem has some meaning if 2^aleph0 > aleph0, but in my first post in this thread i show that (2^aleph0 >= aleph0) = {}.

[Organic]

Master_coda

Fullness is the highest limit of any form of information.

Emptiness is the lowest limit of any form of information.

[master_coda]

Originally posted by Organic
Master_coda

Fullness is the highest limit of any form of information.

Emptiness is the lowest limit of any form of information.

This is exactly the problem with you saying emptiness is {}, and saying that axiomatic set theory depends on it. The mathematical definition of the empty set is the set A such that x\notin A is always true, regardless of what x is.

Your definition talks about "lowest limits of information" which is just something else you haven't defined.

Then you seem to apply your non-mathematical definition to the standard definition in set theory.

[Organic]

Master_coda,

At this point i need your help.


Zf set theory defines the empty set as:

the set A such that x not in A is always true, regardless of what x is.

If you look at this definition, then x must be some existing input form of information.

My idea go deeper then that, and start its research by including the limits of any form of information.


Can you help me to define this idea in a formal way?

Thank you.

Organic

[master_coda]

Originally posted by Organic

the set A such that x not in A is always true, regardless of what x is.

If you look at this definition, then x must be some existing form of information.



Why must x be an existing form of information?

[Organic]

And what if x is Emptiness?

[master_coda]

Originally posted by Organic
And what if x is Emptiness?

Then x is not in the empty set.

[Organic]

And what if x is Fullness?

[master_coda]

Originally posted by Organic
And what if x is Fullness?

Then x is not in the empty set.

No matter what x is, x is not in the empty set.

[Organic]

x is the input A is the set.

the set A such that x not in A is always true, regardless of what x is.

Whet is A if x is Emptiness?

What is A if x is Fullness?

What is A if x is not Epmtiness nor Fullness?

[master_coda]

Originally posted by Organic
x is the input A is the set.

the set A such that x not in A is always true, regardless of what x is.

Whet is A if x is Emptiness?

What is A if x is Fullness?

What is A if x is not Epmtiness nor Fullness?

A is the same thing in all three cases. A is the empty set. It doesn't matter what x is.

[Organic]

A is the same thing in all three cases. A is the empty set. It doesn't matter what x is.

Really ?

x=Eemptiness

the set A such that Emptiness not in A is always true.

[master_coda]

Originally posted by Organic
Really ?

x=Eemptiness

the set A such that Emptiness not in A is always true.

Of course, I'm still waiting for a mathematical definition of Emptiness from you. But however you define it, it isn't contained in the empty set.

[Hurkyl]

Your post is poorly written, my best guess is that this sentence is supposed to be a definition of A:

the set A such that x not in A is always true, regardless of what x is.

And if that is correct, then A has been defined; it doesn't become something different. master_coda would thus be correct.

[Organic]

No, you used a formal definition and learned through our last posts
that this formal definition has logical holes in it that you did not close.

Please show me why do i have to define intuiative concepts like 'Emptiness' and 'Fullness'?

[master_coda]

Originally posted by Organic
No, you used a formal definition and learned through our last posts
that this formal definition has logical holes in it that you did not close.

Please show me why do i have to define an intuiative concepts like 'Emptiness' and 'Fullness'?

Because in math, you have to define everything. Intuitive concepts have no mathematical value until they've been formally defined.


If you use the definition Emptiness=Empty Set then it is still true that \mathrm{Emptiness}\notin\mathrm{Empty Set}.

If you define Emptiness as "the thing that is contained in the empty set" than your formal system is inconsitent.

If you provide a non-mathematical definition than there's no point in even talking about what your definition has to do with math.

[Hurkyl]

Please show me why do i have to define intuiative concepts like 'Emptiness' and 'Fullness'?

You wanted help expressing your ideas in a formal way. That entails defining everything.

[Organic]

There is no such a thing like "Empty set".

All we have is the set concept, and its name is given by its content.

We cannot separate between a set's name and its content's property,
as you wrongly show in your example.

[Organic]

Hi Hurkyl,

Then what is the definition of the set concept?

What is the definition of the content concept?

What is the definition of the number concept?

What is the definition of belonging?

[master_coda]

Originally posted by Organic
There is no such a thing like "Empty set".

All we have is the set concept, and its name is given by its content.

We cannot separate between a set's name and its content's property,
as you wrongly show in your example.

Ultimately, this is why your math is worthless. The name of the set is arbitrary. Call it the empty set. Call it \varnothing. Call it xerfniernisetjilsegtilnerilsneirk. It doesn't matter.

Saying "the empty set should contain emptiness because it's name is the empty set" doesn't mean anything. Math isn't about what you think should be true.

Math is about taking definitions and applying logic to them to see what conclusions you can try. If you aren't willing to define things, than you can't do math. It's as simple as that.

[Hurkyl]

A "set" is an object in ZFC. (or substitute your favorite set theory)

I have no idea what "content" is because that's your idea and you haven't defined it.

As for "number" you're going to have to be more specific; e.g. do you mean real number?

[Organic]

master_coda

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

[Organic]

Hurkyl,


A "set" is an object in ZFC.


x='set'

y='member'

A x is an object in ZFC.
A y is an object in ZFC.

How can i distinguish between them by your definition?

[Hurkyl]

Everything in ZFC is a set, including members of other sets.

(There are other set theories, including some very similar to ZFC, where sets can contain things that aren't sets)

[master_coda]

Originally posted by Organic

x='set'

y='member'

A x is an object in ZFC.
A y is an object in ZFC.

How can i distinguish between them by your definition?

x is a set if it can be constructed using the axioms of ZFC. y is a set if it can be constructed using the axioms of ZFC.

If y is not a set, then x is not equal to y.

If y is a set, then you can determine if x and y are equal using the axiom of extensionality.

Of course, in ZFC everything is a set, so the case of "y is not a set" doesn't actually matter.

[Organic]

So, set is an object in ZFC.

Then what is an object?

[Hurkyl]

Then what is an object?

As used here, it's just a descriptive English word, and not a mathematical term. (Actually, so is "set" in this case, though in, say, Category Theory or NBG "set" is actually a mathematical term)

[Organic]

Hukyl,

Also "set" is just an Enlgish word.

Therefore we are in a circular definition like:

... a set is an object is a set is an object is ...

[Hurkyl]

However, the axioms of ZFC are not just english words; they clearly define what one may do with sets.

[Organic]

Hurkyl,

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

Please tell me what is A?

[master_coda]

Originally posted by Organic
Hurkyl,

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

Please tell me what is A?

A is the empty set. We don't need to know the properties of x, since the definition of A doesn't mention any of the properties of x. If the definition said somewhere "x must have property y" then we would need to know something about x. But the definition doesn't depend on what x is.

[Organic]

master_coda,

The definition is fine, but A's name depends on x property.

So, here is my question again:

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

Please tell me what is A?

[Hurkyl]

\varnothing is defined by:

\forall x: x \notin \varnothing

This is well defined, because one can prove that:


\forall y: \left(
( \forall x: x \notin y ) \Rightarrow y = \varnothing
\right)



IOW if A is a set such that for all x, x \notin A, then A = \varnothing.

[master_coda]

You can name A whatever you want. The name is just an arbitrary label. If you don't like calling it the empty set, then call it whatever you want. Just make it clear that your name for it is a label for the thing mathematicians call the empty set.

There is only one set that satisifies the definition
\forall x\colon(x\notin\varnothing)

[Organic]

So to get A as an Empty set we have to define it like that:

if A is a set such that for all x,x not in A, then A={}(=Empty set) .

But:

Emptiness=

All x=Emptiness

What is set A?

[master_coda]

Originally posted by Organic
But:

Emptiness=

All x=Emptiness

What is set A?

I don't understand what you're asking.

[Organic]

to get A as an Empty set we have to define it like that:

COND='all x' Or COND='any x'

if A is a set such that for COND,x not in A, then A={}(=Empty set) .


Do you agree with both COND?

[master_coda]

Originally posted by Organic
to get A as an Empty set we have to define it like that:

COND='all x' Or COND='any x'

if A is a set such that for COND,x not in A, then A={}(=Empty set) .


Do you agree with both COND?

Yes.

[Organic]

Ok,

So to get A as an Empty set we have to define it like that:

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

Emptiness=

Any x=Emptiness

Then what is set A?

[master_coda]

Ah, I see what you're saying.

If you have no x, then you have nothing. Not even the empty set. There is no A.

That's why the ZFC axioms include the axiom of the empty set. It asserts that the empty set exists.

[Organic]

Bravoooo !!!


And the opposite concept of Emptiness is Fullness.

[master_coda]

But we don't have to worry about not having any x. Because we know that empty set exists.

And I believe its been mentioned before...you can't define things with "opposite". The idea of opposite depends a great deal on context, so without supplying one, you can't use it.

[Organic]

But there is another point of view on x.

Like in a computer program x can be a variable of some value.

Therefore if x= , then x as a variable exists, but without any content.

Emptiness=

Any x=No Emptiness

Now, to get A as an Empty set we have to define it like that:

if A is a set such that for any x-content,x-content not in A, then A={}(=Empty set) .

But:

Emptiness=

Any x=Emptiness

Then what is set A?

Answer: Set A is a non-empty set.

[master_coda]

Why would we want to define the set A in such a way?

Besides, what does it mean for something to exist, but have no content?


I certainly hope you aren't thinking that variables in math are anything like variables on a computer. They're two very different concepts that unfortunately use the same name.

[Organic]

{} exists but has no content.

[master_coda]

Then what is x-content supposed to be?

[Organic]

In the x computer-model x is like a temporary container that delivers its content to the final destination, which is some set.

The deliverd thing is called x-content, which defines set's property.

[master_coda]

You can't really use such a computer model to describe math. Computers have a concept of time. Math doesn't.

Variables in math do not change over time.

[Hurkyl]

to get A as an Empty set we have to define it like that:

COND='all x' Or COND='any x'

if A is a set such that for COND,x not in A, then A={}(=Empty set) .


Do you agree with both COND?

If you're doing literal text substitution, I agree with the statement:

"If A is a set such that for COND, x not in A, then A = {}".

I would like to remark that "all x" is not a grammatical unit, and "for all x" is a quantifier, not a condition.


Emptiness=

As is used throughout mathematics, "=" is a binary relation; it is nonsensical to use it in this way. In order for this statement to have meaning, you are going to have to define it through some means. (axiomatically is fine)


Any x=Emptiness

Similarly, as I mentioned before, "Any x" as used in ordinary logic cannot be used in this way. You are going to have to define this usage through some means as well.

[Organic]

Please look at this:

http://mathworld.wolfram.com/DummyVariable.html


Then instead of the computer model, let us use the dummy-variable model.

[Hurkyl]

In the x computer-model x is like a temporary container that delivers its content to the final destination, which is some set.

You're going to have to define this in some way as well.

[Organic]

Hurkyl,

http://trochim.human.cornell.edu/kb/dedind.htm

As i see it "all x" is a deductive point of view,

and "any x" is an inductive poit of view.


About the x-container please look here:

http://mathworld.wolfram.com/DummyVariable.html

About the dummy-variable as the x-container.

[master_coda]

All math is deductive. Logic is deductive if your conclusions follow necessarily from your premises.

Just saying "x is a dummy variable" doesn't explain what x is.

[Hurkyl]

However, the universal quantifier "for all" has a well-defined meaning.

[phoenixthoth]

hi organic,
you asked us over and over to read your first post and write detailed remarks on it. can you tell us what the flaw in cantor's diagonal argument is to the best of your ability?

theorem: cantor's diagonal argument (one version)
there is no function from any set onto its powerset.

proof: let x be a set. p(x) is its powerset. let f be any function from x to p(x). we will show that f is not onto. to do this requires an element of p(x) (ie a subset of x) that is not mapped to by f. let wf be the well formed fomula (which necessarily depends on the nature of f) that states that "x' is not an element of f(x')", ie, x^{\prime }\notin f\left( x^{\prime }\right). by the axiom of subsets, because wf is a well formed formula, the following object i will define is a set that necessarily depends on the nature of f:

D_{f}:=\left\{ x^{\prime }\in x:w_{f}\left( x^{\prime }\right) \right\} =\left\{ x^{\prime }\in x:x^{\prime }\notin f\left( x^{\prime }\right) \right\} .

i claim that this is an element of p(x) not mapped to by f, which satisfies the requirement and will complete the proof.

to do this, i will argue from \left[ P\rightarrow \left( Q\leftrightarrow \symbol{126}Q\right) \right] \rightarrow \symbol{126}P being a tautology where P and Q are well formed formulae; ie, i will argue by contradiction.

first of all, Df is an element of p(x) because it is a subset, by the subsets axiom, of x.

let P be the statement "f maps an element to Df." in our language, that means \exists x^{*}\in x\left( f\left( x^{*}\right) =D_{f}\right) .

to prove ~P, all i must do, because of the quoted tautology, is prove Q<->~Q assuming P is true. suppose \exists x^{*}\in x\left( f\left( x^{*}\right) =D_{f}\right) . let Q be the statement "the x*, that exists by assuming P, is an element of Df," i.e., x^{*}\in D_{f}. now we show Q->~Q and then ~Q->Q.

Q->~Q: suppose x^{*}\in D_{f}. remember that f\left( x^{*}\right) =D_{f}, so x^{*}\in f\left( x^{*}\right). but since x^{*}\in D_{f}, this by definition means that x^{*}\notin f\left( x^{*}\right) , which the same as ~Q since f\left( x^{*}\right) =D_{f}.

similarly, ~Q implies Q.

QED

comments:
0. i have examined russell's paradox, which involves the same tautology, and cantor's diagonal argument, and i believe i have come up with something.
1. the tautology i quoted is no longer a tautology in fuzzy logic. maybe that's what drove the author royden to intuit that one should eschew all proofs by contradiction...
2. the subsets axiom could be a bad axiom: perhaps sets that lead to contradictions don't exist for consideration.
3. your arguments do not prove that |P(N)|=|N|. however, if you modify the details a bit, you can prove that if U is the absolutely infinite set then not only are P(U) and U in bijection, they are equal! those arguments simply don't work for anything less than U.

read the logic sections under philopsophy in the threads entitled:
1. russell's paradox, the achilles heal of solipsism
2. a new [sic] kind of logic.

please read those before replying.

cheers,
phoenix

[moshek]

Dear phoenix,

Your representation of Cantor theorem is a very well done work.
I am glade that you can see a little similarity to Russell paradox.

I am waiting to Organic replay.

Best,
Moshek[:))]

[Organic]

Hi phoenixthoth,

Please read both parts of:

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

About russell's paradox, please read this:

http://www.geocities.com/complementarytheory/Russell1.pdf

I will read what you ask me to read.

Yours,

Organic

[HallsofIvy]

The fact that you start off http://www.physicsforums.com/showth...60549#post60549
by asserting that 0 is a positive number does not speak well for the rest!

Your argument is based on:

This is not logical but a structural point of view on this paradox, and it is based on the simple fact that there can not be any separation between a set and the properties of its contents.

which is non-sense. You seem to interpret this as meaning "a set must have the same properties as its contents" since you use it to assert that the "set of all sets that do not contain themselves" does not contain itself.

Of course, it's not true that "a set must have the same properties as its contents". The set of all positive integers is not a postive integer! The contents of the empty set have the property that they do not exist. Do you assert that the empty set does not exist?

[moshek]

Organic Hi,

Phoenix read many of your files. Don't sent him now to read more.

It is now your turn to read his very nice representations
to Cantor theorem and tell us what is wrong there, if at all.

Moshek[:))]

[Organic]

Hi HallsofIvy,

The Empty set exists because of an Axiom.

Any set that including the nagation of its content's property = {}.

Also please read this:

http://www.geocities.com/complementarytheory/Russell1.pdf

where i use Z* instead of W.

[Organic]

Dear Moshek,

My answers to phoenixthoth are in these papers:

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

http://www.geocities.com/complementarytheory/Russell1.pdf

I wish you read them too, instead of speaking in the name of phoenixthoth.


Yours,


Organic

[phoenixthoth]

organic,
and we have a proof saying that Boolean Logic cannot
deal with infinitely many objects, in infinitely many magnitudes.


you are correct if you replace "infinitely many objects" with U, the universal set of absolute infinity.

also look up inaccessible cardinals and such things for other objects that are also "really big" that are hard to prove exist as far as i know.

we're really thinking about the same thing but boolean logic can prove that there is no map from x onto p(x) EXCEPT for x=U, the universal set which not only is in bijection with p(U), U=p(U), which is a stronger statement than you say about N, which if i remember right, was something like |N|=|P(N)|. i am about to read your russell's paradox paper and i bet i'm going to find things that are on the right track as well.

ever heard of the hundreth monkey syndrome? it corresponds to the conjecture that often "discoveries" happen independently simultaneously. what wasn't clear to me when i first read your combinations article was that your arguments make more sense if whenever you wrote about N, you were actually writing about the universal set U.

we can say that U can exist axiomatically from the perspective of three valued logic and then stick to two valued logic for the rest in an effort to use two valued logic as much as possible. but your statement that i quoted says this exact thing: boolean logic cannot handle U. kudos to you, organic!

All those kinds of questions are meaningless questions, and they do not lead to any
paradox.

that's exactly what it could mean when a logic statement has the third truth value. this is when mu is the answer. now if you look carefully, you see that it's not U that has meaningless statements surrounding it, it's russell's subset that has meaningless questions surrounding it. i'd also like to point out that cantor's diagonal argument that in 2-valued logic proves that for x!=U, there is no function from x to p(x), and the Df referred to is precisely the set you get in russell's paradox when you consider the identity function from x to p(x) (i think). so the same resolution of the paradox, the appeal to 3 valued logic, applies to show that U is in bijection with p(U). later, i showed that any set in bijection with U is U, hence U=p(U). in fact, any time there is a 1-1 function from U to x, then U=x.

check out the scattered remains on my discussion forum for the search for absolute infinity:
http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=129;st=20;r=1;&#entry573
we were really on the same hunt with combinations and this search.

if only cantor considered using 3 valued logic, we wouldn't be discussing this right now, most likely. his "crippling" attachment to 2 valued logic along with his unwillingness to see past the paradox can definitely drive one mad. i'm not saying we now move to 3 valued logic. all i suggest is that 3valued logic implies the universal set can be axiomatized into existence. furthermore, if that article by max tegmark on the theory of everything is correct, and there are self-aware structures, i postulate that not only is U such a self-aware structure (by no means the smallest SAS), that it, in some weak sense, is omniscient at least of all SAS's that are sets under the supposition that all sets are aware at least weakly of their contents as well as some form of awareness of all sets with nonempty intersection with them and perhaps something also to do with sets they can be mapped onto. i'm just taking a shot in the dark, but i'm guessing that the level of self awareness is in some relation to it's cardinal number. well, if that's true, then U would be the most aware set. however, in that sense, categories would probably be more self-aware and the category of all categories might reign supreme in the self-awareness book. i really don't know what self-awareness is nor how long it will take to figure out what makes us self-aware but for now perhaps if we postulate that we get self-awareness somehow from being in U (note that any manifold such as the one our physical universe is in must be a subset of U), which if any sets have self-awareness implies perhaps that U does, too.

i can't stress highly enough that 2 valued, boolean, logic is just fine for sets that aren't weird subsets of U or U itself.

conjectures on SAS's: http://207.70.190.98/scgi-bin/ikonboard.cgi?act=ST&f=2&t=196&st=&&#entry574

[Organic]

HallsofIvy,

To get A as an Empty set we have to define it like that:

x is something

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

x is nothing

if A is a set such that for any x,x not in A, then A=a Non-empty set .

As you can see the property of A depends on the property of x.


The set of all positive integers is not a postive integer


We cannot use the word 'all' together with 'infinitely many objects'
because the basic property of a set with 'infinitely many objects'is not to be completed.

We can use the words 'all' or 'complete' only if we can reach all of their objects, which means, only with sets that have 'finitely many objects'.

This is a fundamental change in the meaning of infinity in Math language.

Please read this:

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

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



Organic

[phoenixthoth]

To get A as an Empty set we have to define it like that:

x is something

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

x is nothing
correction: x is nothing in particular.

As you can see the property of A depends on the property of x. no, it doesn't. what properties does it depend on?



We cannot use the word 'all' together with 'infinitely many objects'
because the basic property of a set with 'infinitely many objects'is not to be completed.
the universal set is completed. what you're saying works for U but not for N.

We can use the words 'all' or 'complete' only if we can reach all of their objects, which means, only with sets that have 'finitely many objects'.

This is a fundamental change in the meaning of infinity in Math language.
yes, it is. |U|=OMEGA

on my alephnull website, i proved that U=P(U), remember, so that U really does have the largest cardinal.

[Organic]

Hi phoenixthoth,


I am going to read your post (the one with the web sites) and only then I'll reply to you.

Thank you for your detailed reply.


Yours,


Organic

[phoenixthoth]

i'm waiting to also see what hurkyl thinks.

[Organic]

correction: x is nothing in particular.

'nothing in particular' is the same as if i say that 'x is something' XOR 'x is nothing'.


Therefore A depends on the property of x.

The axiom of the empty(XOR non-empty) set:

if A is a set such that for any x,x not in A, then A=(depends on the property of x, which can be at least something XOR nothing).


Organic

[phoenixthoth]

fine. then it is something, but nothing in particular*. however, it does not depend on properties of x. what properties does it depend on? what are you really talking about? Ø, U, or something else? are you saying that it's all basically meaningless? well, from one perspective, it is. they're just symbols. even {} is a meaningless symbol and to lots of people, OMEGA is a meaningless symbol. but if you realize what others have realized, then you can see the meaning.

*duality

[Organic]

I am talking about the limits of any information system, including Math language.

Please read this:

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

and this:

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


Thank you.


Organic

[Hurkyl]

I think that if you're going to talk about a universal set, you're going to need to explicitly demonstrate why the usual proof of X \neq \mathcal{P}(X) fails, and how the usual constructions of paradoxes fails. (such as the set of all sets that don't contain themselves)

[Hurkyl]

If "X = nothing", does that mean X \in \varnothing, X = \varnothing, or something else?

Is it the case that \forall a : a \in \{\_\}?

[Organic]

x is the property of some set.

The most basic properties are: Empty XOR Non-empty set.

[chroot]

Organic,

It appears that even after 15 pages of patient guidance, you continue to use your own ill-formed, imprecise, and ambiguous notation, such as "fullness = {___}" as if no one had ever objected to it.

Haven't you learned anything from these people?

- Warren

[Hurkyl]

Is "Nothing" is a logical predicate? I.E. "X is nothing iff X = \varnothing"?


Is it the case that \forall a : a \in \{\_\}?

[Organic]

Hi chroot,

Please read this:

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

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

and then please reply.

Thank you.


Orgainc

[Organic]

Hurkyl,

0) There is only one form of nothing, which is the "content" of the Empty set ( {} ).

There are at least three forms of something:

1) finitly many objects ( {a,b} ).

2) infinitely many objects ( {a,b,c,...} ).

3) Fullness ( {__} ).


(0) and (3) are Actual infinity (the lowest and highest limits of any information system).

(2) is potential infinity (cannot be completed).

Please read this:

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

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

Thank you.


Organic

[Hurkyl]

Does "content" have anything to do with \in?

Is it the case that \forall a : a \in \{\_\}?

[Organic]

Content is what gives to some set its property(ies).

[Hurkyl]

What is a property of a set?

Is it the case that \forall a : a \in \{\_\}?

[phoenixthoth]

Originally posted by Hurkyl
I think that if you're going to talk about a universal set, you're going to need to explicitly demonstrate why the usual proof of X \neq \mathcal{P}(X) fails, and how the usual constructions of paradoxes fails. (such as the set of all sets that don't contain themselves)

well, that the set of sets that don't contain themselves is russell's paradox. the point is that, as i argued in the thread "russell's paradox: the achilles heal of solopsism," if you use 3-valued logic, then the tautology used to argue by contradiction is no longer a tautology. furthermore, a way around 3-valued logic is to state that in the subsets axiom, if a well formed formula leads to a contradiction, then that is not a subset.

let U be the universal set. i will show how cantor's diagonal argument fails to show that U does not have a map onto P(U). let id_{U} be the identity map on U; i will show that id_{U} maps onto P\left( U \right) if contradiction is not a tautology (which it isn't in 3-valued logic).

first of all, note that since \forall x\left( x\in U\right) , it follows that if x is a set then x\subset U, hence P\left( U\right) \subset U so it makes sense to think of id_{U} as having P\left( U\right) involved in the range.

let's follow cantor's diagonal argument, applied to id_{U}.

D_{id_{U}}:=\left\{ x\in U:x\notin id_{U}\left( x\right) \right\} . but by what id_{U} is, this means that D_{id_{u}}=\left\{ x\in U:x\notin x\right\} . note that D_{id_{U}}\in P\left( U\right) . then if id_{U}
is onto, id_{U}\left( x^{\ast }\right) =D_{id_{U}} for some x^{\ast }\in U. hence, x^{\ast }=\left\{ x\in U:x\notin x\right\} . now we have a contradiction in two valued logic: x^{\ast }\in x^{\ast }\leftrightarrow x^{\ast }\notin x^{\ast }.

this argument is based on a tautology called "argument by contradiction" and this tautology is no longer a tautology in 3-valued logic.

one can show that if there is a 1-1 map from U to x, then U=x. hence, U=P\left( U\right) . also, if there is a function from x onto U
, then x=U.

[Hurkyl]

In what I have read (long ago) on multivalued logics, the classical paradoxes in binary logic can fairly straightforwardly be extended to multi-valued logic.

For instance, S := \{ x | x \in x\; \mbox{is not true} \} sufficies for at least one ternary logic.


if a well formed formula leads to a contradiction, then that is not a subset.

And there's the rub; we need to know what a "safe" set of formulas is. This is a purely metamathematical concern; I can't see any way it could be written formally.


Anyways, it is an interesting exercise to formally write up your ternary logic and see if it really sufficies. I bet that replacing "p \notin q" in Cantor's argument with "p \in q is false or <name of third logical value>", then you can still derive a contradiction.

(actually, my gut tells me that you don't even need this modification; but I can't be sure until I see just what your ternary logic looks like)

[moshek]

6. Mathematical treatment of the axioms of physics
The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

As to the axioms of the theory of probabilities,14 it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.

Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach,15 Hertz,16 Boltzmann17 and Volkmann. 18 It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Thus Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely one might try to derive the laws of the motion of rigid bodies by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. For the question as to the equivalence of different systems of axioms is always of great theoretical interest.

If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. At the same time Lie's a principle of subdivision can perhaps be derived from profound theory of infinite transformation groups. The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed.

Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain such proof always forces us most effectually to an exact formulation of the axioms.

David Hilbert( Paris 1900)

[Organic]

General Information Framework (GIF) set theory


Set (which notated by ‘{‘ and ‘}’) is an object that used as General Information Framework.

Set's property depends on its information’s type.

There are 2 basic types of information that can be examined through GIF.

1) Empty set ={}
2) Non-empty set

(2) has 3 non-empty set’s types:

1) Finitely many objects ( {a,b} ).
2) Infinitely many objects ( {a,b,…} ).
3) Infinite object ( {__} }.

(3) Is the opposite of the Empty set, therefore {__}=Full set.

GIF has two limits:

The lowest limit is {}(=Empty set).

The highest limit is {__}(=Full set).

Both limits are unreachable by (1) or (2) non-empty set’s types.

Infinitely many objects ( {a,b,…} ) cannot be completed, therefore words like ‘all’ or ‘complete’ cannot be used with sets that have infinitely many objects.

{} or {__} are actual infinity.

{a,b,...} is potential infinity.

An example:

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

[master_coda]

Originally posted by Organic
General Information Framework (GIF) set theory

So what can we do with this theory that we can't do with standard set theory?

[phoenixthoth]

In what I have read (long ago) on multivalued logics, the classical paradoxes in binary logic can fairly straightforwardly be extended to multi-valued logic.

For instance, S := \{ x | x \in x\; \mbox{is not true} \} sufficies for at least one ternary logic.

i'm going to have to look at this further. since you replaced it with the word false, i'd just like to say that things are not exclusively true or false anymore but possibly the third truth value. just for everyone else, i'll put what i know about 3 valued logic here:
let 0 mean F, .5 mean the third truth value M, and 1 mean T.
V(P) gives the truth value of the property (aka well formed formula) P. to use "max" below, we could just say F < M < T.

V(AvB)=V(BvA)=max{V(A),V(B)}.
V(~A)=1-V(A).

(in this language, your S is S:=\left\{ x\in U:V\left( x\in x\right) \neq T\right\} and i'll think about this. is that something you can state in 1st order logic as the subsets axioms is stated?)

from this, one can derive the truth tables for the other truth values using the rules A^B=~(~A v ~B), A->B = ~A v B, and <-> is what it usually is.

apparantly, there are 3072 3 valued logics but i doubt all of those are generalizations of 2 valued logic.

note that V can be any function that generalizes 2valued logic and this one does. let me now write out some truth tables, ending in the truth table for the argument by contradiction, which is \left[ A\rightarrow \left( B\leftrightarrow \symbol{126}B\right) \right] \rightarrow \symbol{126}A. the first two columns will be the truth values of A and ~A. then there will be a double bar ||. the next truth values will be B and ~B, followed by B->~B and ~B->B. the next pair will be B<->~B and A->(B<->~B). the final truth value will be ~A. the main result is that the final values are not always T; ie, this argument is not applicable because it's not a tautology anymore.
1.TF||TF||FT||FF||T
2.TF||MM||MM||MM||M
3.TF||FT||TF||FF||T
4.MM||TF||FT||FM||M
5.MM||MM||MM||MM||M
6.MM||FT||TF||FM||M
7.FT||TF||FT||FT||T
8.FT||MM||MM||MT||T
9.FT||FT||TF||FT||T

the standard russell's paradox is to use the following:
A states the universal set exists (normally, ~A is T)
B states that S &isin; S.


and there's the rub; we need to know what a "safe" set of formulas is. This is a purely metamathematical concern; I can't see any way it could be written formally.
it would require quantifying over wffs as far as i can see. this would be added somewhere:
if there is a wff such that it implies a contradiction, then there is no subset with that wff as a defining property.
yes, this would involve an investigation of "safe" wffs. we would only have to try this if one were to not allow 3-valued logic to get around what russell's paradox is currently.

there's no sign that some other paradox won't rule U out; if there are others, do they also rely on contradiction which is no longer a tautology in 3 valued logic? i'm not suggesting we use 3 valued logic for everything. it is a generalization of 2 valued logic used only when neccessary such as to say things like "from one perspective, russell's theorem is a nontautology."



Anyways, it is an interesting exercise to formally write up your ternary logic and see if it really sufficies. I bet that replacing "p \notin q" in Cantor's argument with "p \in q is false or <name of third logical value>", then you can still derive a contradiction.
i'll look at your S. thanks for submitting it.

organic,
i think what i've been doing is what you're after but it only applies to the universal set, not just every infinite set. you said it, in spirit: binary logic cannot handle infinitely many objects. while that's technically incorrect, it is definitely true of the absolute infinity. and i think i can show how 3 valued logic can handle russell's paradox.

for anyone interested, the stanford encyclopedia has nice articles on many-valued logic:
http://plato.stanford.edu/entries/logic-manyvalued/
there it gives the possibility of using it to resolve some paradoxes but it doesn't mention russell's from what i saw.

[Organic]

Hi master_coda,

Good question:

Please look at this two articles of mine:

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

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


At this stage you have to look at them as non-formal overviews, but with a little help from my friends, they are going to be addressed in a rigorous formal way.

All the energy that was used to research the transfinite universes, is going to be used to research the information concept itself, including researches that explore our own cognition's abilities to create and develop the Math language itself.

By GIF set theory our models does not have to be quantified before we can deal with them, because GIF set theory has the ability to deal with any information structure in a direct way, which keeps its dynamic natural complexity during the research.

Concepts like symmetry-degree, Information's clarity-degree, uncertainty and redundancy, are some of the fundamentals of this theory.


Organic

[phoenixthoth]

hurkyl,

define S so that S=\left\{ x\in U:V\left( x\in x\right) \neq T\right\} .

let P be the statement S &isin; S and Q be the statement V\left( S\in S\right) \neq T.

in the following truth table, i'm using the same rules as above and the first column is the truth value of P, the second column is the truth value of Q, and the third is the truth value of Q<->~Q:
1. T||F||F
2. M||T||M
3. F||T||F

note that it's not always F. in fact, it's M when V\left( S\in S\right) = M. thus, one possible conclusion is that if "undecidable" means neither true nor false, then it is undecidable whether an undecidable statement implies Q<->~Q.

hence, one can axiomatize it either way. S &isin; S has truth value M does not lead to a contradiction. in other words, it is "unknown" whether S &isin; S, if to know something means to know it is either true or false.

however, i'd like to avoid attaching meaning to M yet. just arguing syntactically, not semantically based on the meaning of T, M, or F. syntactically, this is a valid system and the contradiction is no longer tautologically false. thus, it appears to me that this expanded paradox is not a paradox in 3 valued logic.

[Hurkyl]

The thing is, it seems, Cantor's argument proves true this statement:

S \in S \Leftrightarrow \neg(S \in S)

However, no truth value assignment to S \in S permits this statement to be true.



I made this truth table, but it turns out I didn't need it for this post. However, I'm leaving it for future reference. [:)]


\begin{array}{c | c | c | c | c | c | c}
A & B & A \vee B & A \wedge B & \neg A & A \Rightarrow B & A \Leftrightarrow B \\
\hline
T & T & T & T & F & T & T \\
T & M & T & M & F & M & M \\
T & F & T & F & F & F & F \\
M & T & T & M & M & T & M \\
M & M & M & M & M & M & M \\
M & F & M & F & M & M & M\\
F & T & T & F & T & T & F \\
F & M & M & F & T & T & M \\
F & F & F & F & T & T & T
\end{array}

[Hurkyl]

It occurs to me that ternary logic has some annoying problems... for instance, A \Leftrightarrow A is not a tautology. It seems one can no longer use A \Rightarrow B to be synonymous with "If A then B", and one can no longer use A \Leftrightarrow B to be synonymous with "A and B are the same"...

[phoenixthoth]

thanks hurkyl. would it be possible to just use ternary logic to say that russell's paradox is based on a nontautology, axiomatize the universal set into existence that way, look for other paradoxes that ternary logic can't handle, and then stick to binary logic for the rest of set theory? if one ever uses ternary, are they bound to always use it, kind of a consistency thing from an esthetic viewpoint vs consistency from a strictly mathematical viewpoint?

i would like to point out that A<->A is a tautology if V(A)!=M. since nothing else in set theory requires ternary logic, it seems like nothing is harmed.

i was also thinking of using a lattice-theoretic approach to set theory in which Ø and U are axiomatized into the system, along with the usual other sets. some problems:
1. how to define &isin;
2. how to define subset
3. how to reformulate the subsets axiom
4. avoid russell's paradox and ternary logic.

the thing is, i bet people have tried all this before at some point but since i don't know about it, it probably either all failed or never caught on.

[Hurkyl]

I've never seen a non-binary logic presented at a foundational level; I can't say what's a "normal" thing to do.


Anyways, an approach that works for general purpose is to strike out the axiom of the power set; you only work in an explicitly specified "universe" (which is not the universe of all sets). For instance, one can (I believe) do the whole of analysis in a thing called a "superstructure" which is constructed as:


\mathcal{S}_i = \left\{
\begin{array}{l l}
\mathbb{R} \quad & i = 0 \\
\mathcal{S}_{i-1} \cup \mathcal{P}(\mathcal{S}_{i-1}) & i > 0
\end{array}
\right


and the superstructure is defined to be \mathcal{S} = \bigcup_{\iota \in \mathbb{N}} \mathcal{S}_{\iota}.

So in the superstructure, you can indirectly find the power set of many sets by appealing to the structure of the superstructure, but you aren't allowed to take the power set of an arbitrary set.


(note that \mathcal{S} is a lattice, if you use union for join and intersection for meet... or is it the other way around?)

[phoenixthoth]

you can further that structure if it's not large enough by replacing R with S.

i always wondered why my set theory teacher told me some set theorists don't like the powerset axiom...

[Hurkyl]

Right. (and on a side note, if you replace \mathbb{R} with \mathbb{N}, you get the same thing)


But the thing is, it is large enough, so you don't need to do allow anything recursive to occur. [:)]


Intuitively, all of the bad things start happening when you allow sets to get "too big". The power set axiom is, I believe, the only way to make big sets in ZFC, so if you get rid of it, things won't get too big!

[phoenixthoth]

there is no upper "limit" to this, is there?

more generally, a universal set could never be constructed from union and powerset can it? i don't think it can because the powerset is always bigger. even some kind of transfinite recursion, if such a thing exists, would have to be of a nonabsolute infinitary nature thus not big enough to construct U "from below."

[Hurkyl]

Right, the power set is always bigger. If you could prove the existance of a universal set, that would be a contradiction in ZFC.


There is such a thing as transfinite recursion, but it's tricky to set up... I don't see how you could even think about pointing towards a universal set, though; the two biggest problems I see are:

1) ZFC permits sets that cannot be constructed from its axioms.
2) The index "set" would be "too big" to be a set

[phoenixthoth]

hurkyl: If you could prove the existance of a universal set, that would be a contradiction in ZFC.

using binary logic. so if i continue working on the assumption that it's ok to use ternary logic once, to climb over the proof by contradiction to show that the universal set can't exist, i would then kind of be a math rebel? rebel or just plain a waste of time? i'm sure someone must have tried this before and i haven't heard of it so it must have failed, right? will you be the dedekind to my cantor? ;)

[Hurkyl]

Ok, spend a few years developing ternary logic and set theory, then I'll show you the flaw. [;)]

[Hurkyl]

But more seriously, you can't just use ternary logic once; if the theory is written in binary logic, you can't introduce ternary logic where you fancy it; you have to either start with ternary logic from the beginning, or use binary logic to model a theory which uses ternary logic.

[phoenixthoth]

do you promise? can i show it to you as i develop it instead of after 3 years?

ok, that's something i had been wondering. i have to recheck what few proofs i have and learn how to prove via ternary logic.

[Hurkyl]

It would certainly be interesting to look at; I'll give you fair warning, though, I'm lazy so I might not be up to doing a lot of research for you. [:)]

[phoenixthoth]

what do you think of the following general plan:
revise all axioms so that whenever there is a well-formed formula W appearing in the axiom that is meant to be true, replace it with
V(W)=T.

for example, if we want A<->B to mean what it usually does, we can say V(A<->B)=T instead of just A<->B. however, if V(A) and V(B) are both M, then V(A<->B)!=T, though that might be a very good thing. so, in other words, replacing A<->B with V(A<->B)=T might be just what i need.

in the axiom of extensionality, there's a statement of the form
\forall x\left( x\in a\leftrightarrow x\in b\right) \rightarrow a=b. in fuzzy logic, these connectives don't mean what they do in binary logic; so let's see what happens when i modify it to this:
\forall x\left( V\left( x\in a\leftrightarrow x\in b\right) =T\right) \rightarrow a=b.

the subtlety is whether or not binary logic must be used when i make a statement like V\left( x\in a\leftrightarrow x\in b\right) =T. with equality, i want it to either be = or !=. that doesn't seem to be a problem. equality is binary whereas the truth value of what's inside the V( ) can be T, M, or F. how does all that sound before i continue?

what i'd like to do is show that this axiom is equivalent to the original axiom. i'll have to think about this.

the main sticking point will be the subsets axiom. how about this:
SS 2. \exists x\forall yV\left( \left( y\in x\leftrightarrow y\in a\wedge A\left( y\right) \right) \right) =T?
this subsets axiom does contradict the universal set axiom.

SS 3. \exists x\forall yV\left( \left( y\in x\leftrightarrow y\in a\wedge A\left( y\right) \right) \right) \neq F.
SS3 does not contradict the universal axiom and V(S&isin;S)=M follows. thus, S could be called a fuzzy subset of U.

the hope is that if the subsets axiom is the only one with any structural difference between it and the original axiom, changing a well formed formula W into V(W)=T in all cases except the SS axiom, in that case changing it to V(W)!=F, then ternary logic will work well.

[Hurkyl]

I think it might be better to start with ternary logic (e.g. what are the rules of deduction... do we have rules of deduction only for T, or for both T and M? that sort of thing), then move onto ternary ZFC... or maybe a simpler theory first.

Doing it in its own thread might be nice too [:)]

[phoenixthoth]

Infinitely many objects ( {a,b,…} ) cannot be completed, therefore words like ‘all’ or ‘complete’ cannot be used with sets that have infinitely many objects.
i agree that complete can't, but why "all?" the quantifier is as in "all sets". why can we not use this? the "therefore" seems to be a non sequitor: the conclusion doesn't follow from the premise.

saying the empty set is {} is meaningless. you have to define it and give it properties such as for all sets x x is not an element of {}.

saying the (absolutely) infinite set {__} without properties is also meaningless. you have to say that for all sets x x is an element of {}. in your theory, you also have to show how russell's paradox is avoided.

[Organic]

Hi phoenixthoth,

Please let's continue here:

http://www.physicsforums.com/showthread.php?s=&threadid=11440


Thank you.


Yours,


Organic

[moshek]

Organic,

Well, You are trying to develop here some non Euclidean
mathematics, so exact definition at the beginning is not the point.

but better you read what happande to Hipasus:

http://www.anselm.edu/homepage/dbanach/pyth3.htm

Happy new year
Moshek

[:))]

[Organic]

Hi Moshek,

Can you show some fundamental differences between Euclidean and Non-Euclidean mathematics?

[phoenixthoth]

one difference is that the sum of interior angles in a triangle is not 180 degrees.

an axiom of euclidean geometry is that given a line and a point not on the line, there is 1 line passing through the point that is parallel to the given line. in non euclidean geometry, this can be replaced with a number other than 1.

[moshek]

It will be very difficult organic because it is so new that it is almost does not exist. Only in the mind of maybe ~10 people in the world today but they are not a community like the big Euclidean mathematics community that exist more that 2000 year's and work in Plato paradigm.

very interesting also is this paper:

http://www.american.edu/academic.depts/cas/mathstat/People/kalman/pdffiles/irrat.pdf


Moshek
[:))]

[moshek]

Hi phoenixthoth,

you are talking about non Euclidean geometry:

you can read that:


http://www.gurus.com/dougdeb/Essays/Geometry/geometry.html

take care

Moshek
[:))]

[Organic]

I have found this opinion, by Prof. Doron Zeilberger, on Non-Euclidean Mathematics:

http://www.math.rutgers.edu/~zeilberg/Opinion43.html

And also this: http://www.math.rutgers.edu/~zeilberg/PG/Introduction.html

What do you think?

[moshek]

Well it look that Z

is one of the 10.

Thank you Organic.

What do you think?

[zz)]

[Organic]

Once upon a time a little fish asked his mother: "Mammy, one of my friends told me the that there exist somthing, which its name is 'water', so Mammy where can we find this water?"

Without another point of view on something, it is hard to understand it.

Please read this: http://www.math.rutgers.edu/~zeilberg/PG/Introduction.html

by Prof. Doron Zeilberger.

[moshek]

I will be honest with you Organic,
I know already Doron and i like his work very much!

I told him that i do not agree
with his direction for the new mathematics.

Computer are only the justification
to create a new mathematics.

What we have to do is the opposite direction
that Doron sagest us here.


Just Look on this point.

.



Moshek
[:))]

[Organic]

What is the opposite direction of Prof. Zeilberger?

[moshek]

Dear Organic,

I will try answer you,
and you are one of the 10
Altow you don’t know at all !!
Euclidian mathematics.

But before I do that
please tell me if you don’t mind,
have you been already
in the forest of the monkeys?

Yours
Moshek
[:))]

[Organic]

What is the forest of the monkeys?

Is it a mathematical slang?

[moshek]

If you come to that so high point in mathematics,
(and you also look like a monkey)
I can't believe that you don’t know "The answer".

[Organic]

From an organic point of view, we never left the forest of the monkeys, because all complexed organism on this planet are DNA products 3 to 4 billion years old.

[moshek]

Organic,

This is not so good answer, ( like pupil to his teacher...)
I will come back in few day's to your 2 questions.

Maybe you will have until than "The answer"

untill thay you will enjoy to look on:

http://www008.upp.so-net.ne.jp/gps1999



Your's
Moshek
[:))]

[Organic]

In my opinion, there is no "The Answer".

[moshek]

ok

[;)]

[moshek]

Organic:

here you can find what is the forest of the monkey


The Book: Mount Analogue ( 1959)


René Daumal’s is a twentieth century classic, combining the author’s poetic gifts and philosophical accomplishments in a manner that is both entertaining to read and profound to contemplate. Among other things, this is a marvelous tale in which the narrator/author, one of an intrepid company of eight, sets sail in the yacht Impossible to search for Mount Analogue, the solid, geographically located, albeit hidden, peak that reaches inexorably towards heaven—as Mount Olympus reached to the home of the Greek gods, or Mount Sinai to the presence of Yahweh. Daumal, often described as one of the most gifted literary figures in twentieth-century France, died before the novel was completed, providing an uncanny one-way quality to the journey.


About the Author
René Daumal (1908-44), a follower of the teachings of G.I. Gurdjieff, also studied Sanskrit, philosophy, science, mathematics, and medicine. He was an editor of the French poetry and surrealist review Le Grand Jeu, and the novel Mount Analogue was first published posthumously, in 1952

[moshek]

Organic

The forest of the monkeys is the base of the Mount Analogue.
I am sory that i take in advance that you know about this place.

Moshek

[Organic]

Then, I repeat:

From an organic point of view, we never left the forest of the monkeys, because all complex organism on this planet are DNA products 3 to 4 billion years old.

The DNA is the base of this mountain, and the fulfillment of the uniqueness of each complex system (based on DNA) is its private peek.

Shortly speaking, DNA principle is general (or maybe global) but its fulfillment is unique (or maybe local).

Global and Local are complementary concepts of Mount Analogue.


Yours,

Organic

[phoenixthoth]

what if global=local somehow? what i mean is that what if the "opposite" ways of looking at mount analogue are complementary but both inadequate so that a transcendence of the opposites is needed in order to really start up that mountain. rather than see things dualistically as global and local, perhaps there is a way to look at it ONE way that is the RIGHT way. but will doing that be the equivalnt of reaching the base of the mountain, or reaching its peak? i think its base. the peak is not something i can imagine yet.

[Organic]

Hi phoenixthoth,


What you call ONE is the BALANCE that exists between opposite concepts, giving them the chance to complement each other instead of destroying each other, when they are meeting.

So, the minimal condition for any thinkable system must be at list 0=x-x where the left size (notated as 0) is the balance and the right side is the opposite's communication environment (notated as x-x).

Please pay attention that from this model, (the 0 result of x-x=mutual destruction) XOR (the 0 result of x-x=mutual communication).


Maybe the Meta system is:

(the 0 result of x-x = mutual destruction)
? = XOR
(the 0 result of x-x = mutual communication)

where the answer to ? is given by self-aware systems.



Yours,

Organic

[phoenixthoth]

seems like destruction balances communication in this theory. what if those are the same thing in that sometimes communication is or results in destruction? what i mean by that is the following example:
an intense frequency penetrates a solid object and it shatters.

maybe it's best to not name certain things:
0=x-x is ?

this idea is expressed in group theory that everything (in a group) has an opposite that cancels it.

it also mentions the idea of IDENTITY: 0=0 yet, in a way, 0!=0 because x-x has more information in it than just 0 yet x-x=0 which results in information loss aka destruction through communication.

[moshek]

Dear Organic and Phoenix:

For being in the forest of the monkey
first you must understand
that we are only the monkey of Euclid.

Yes, "global=local" is a key point to start the travel.
in the sence of it duality on mathematics as a whole.
But if we will work very hard together as a tim
we may start the travel to mount Analogue not before 2006.

The pick is the non-Euclidean mathematics
that non of us can see it today,
because it is not exist yet
it is Only a potential now.

Interesting work:

http://modular.fas.harvard.edu/sga/from_grothendieck.pdf

Moshek



[:))]

[phoenixthoth]

why do you say 2006?

the two concepts for which global=local are Ø and any singleton which is representable as a point. some people conjecture that somehow consciousness has a dual nature: point-like and (with an infinite distribution)-like. perhaps this relates to self aware structures? perhaps global analysis on on part of the dualaity is local analysis on the other part.

[moshek]

http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=6;t=217;st=0;&#entry732

Thank's you for that information phoenix.

[:))]

[Organic]

Hi,

If the Meta system is:

(the 0 result of x-x = mutual destruction)
? = XOR
(the 0 result of x-x = mutual communication)

where the answer to ? is given by self-aware systems, then maybe this is the deep meaning of any open system, which means:

result = ? (where ? is a legal answer)

From the above we maybe can give a moral interpretation to Math results, given by self-aware systems.

[phoenixthoth]

that kind of sounds like reinventing the wheel in terms of godel's incompleteness theorems and undecidability: things where you can prove ? is the answer, roughly speaking. i don't know, perhaps the existence of self-aware structures will be ? within the other known structures so one can express their free will in order to believe they exist or don't.

[moshek]

Well phoenix,

In 1823 Bolyai and Lobachevsky's
without knowing one about the other
invent the Non-Euclidean geometry.
But most of it was already known to Gauss.

In 2006 It may be declare by the IMC in Madrid
who is the new Gauss that can lead us to the
Mount of Analogue , A Non-Euclidean mathematics.


[:))]

[Organic]

I think you miss the point.

1) phoenixthoth is self-aware system.

2) phoenixthoth has the ability to decide if he a destroyer XOR communicator.

3) phoenixthoth decisions has an influence on itself and its environment.

4) phoenixthoth responsibility as a participator is extremely important.

[phoenixthoth]

Originally posted by Organic
I think you miss the point.

1) phoenixthoth is self-aware system.

2) phoenixthoth has the ability to decide if he a destructor XOR communicator.

3) phoenixthoth decisions has an influence on itself and its environment.

4) phoenixthoth responsibility as a participator is extremely important.

1) embedded in a larger SAS

2) i don't think it's xor. let me try to explain. a communicator can be a destroyer at the same time yet destruction is the intention of the one communicated TO not the one doing the communicating. like a high intensity sound wave shattering a glass. it is the glass' intention to be a destroyer from a certain point of view, all the sound wave wants to do is communicate but the power is overwhelming unintentionally. now i think this is expressed in the information loss, also known as destruction, in the equation x-x=0 or 0=0. those seem to be different equations because x-x has more information that has been destroyed AND communicated from x to x in a self-awareness sense.

3) yes and vice versa.

4)yes, as is the responsibility of all SAS's.

[moshek]

824 mathematition already visit :

http://www.icm2006.org/#

[:))]

[Organic]

phoenixthoth,


About a moral interpretation to Math results.

If the Meta system is:

(the 0 result of x-x = mutual destruction)
? = XOR
(the 0 result of x-x = mutual communication)

where the answer to ? is given by self-aware systems, then maybe this is the deep meaning of 'Right and Worng'.

It is good to be a (sum=commounicator), it is bad to be a (sum=destroyer).

Therefore, (0=mutual destruction) XOR (0=mutual communication).

[phoenixthoth]

yes i think that does give a deeper meaning of right and wrong but keep in mind it also depends very much on your perspective. from one perspective, x-x is mutual destroy and from another x-x is mutual communicate. since infinity-infinity is not "always" zero, perhaps that object does not "always" destroy itself, if that makes any sense.

[phoenixthoth]

destruction element= - = communication element

[moshek]

Yes, and the different between Euclidam mathematics
and non euclidian mathematic is almost invisibal.

see:


http://elib.zib.de/pub/Gauss/gauss-pressrelease.htm

Moshek
[:))]

[Organic]

Let us examine the meaning of 0, as a result of x-x.

Meaning 1) 0 is the result x and –x mutual destruction.

Meaning 2) 0 is the result x and –x mutual communication.

There is no sum (by quantity) to finitely or Infinitely many objects when sum is a quality value like destruction XOR communication, because the sum can be the result of a very fine change (the butterfly effect) in the input, which can upside down the whole picture.

In my opinion, this is the deep meaning of a non-Euclidian Mathematics, which is used by self-aware systems to get moral results.

About a moral interpretation to Math results.

If the Meta system is:

(the 0 result of x-x = mutual destruction)
? = XOR
(the 0 result of x-x = mutual communication)

(where the answer to ? is given by self-aware systems) then maybe this is the deep meaning of "choosing between 'Right' and 'Wrong'".

It is 'good' to be a (sum=communicator), it is 'bad' to be a (sum=destroyer).

Therefore, (0=mutual destruction) XOR (0=mutual communication).

'Wrong' is (0=mutual destruction).

'Right' is (0=mutual communication).

In other words, Complementary Logic is the logic of mutual communication between opposite things.

Form this point of view, please look again at:

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

http://www.geocities.com/complementarytheory/4BPM.pdf

[moshek]

Thank you Organic,
for this interesting opinion,
I think that only by
really open mind dialog
we can find the way to see
Non-Euclidian Mathematics

There is no need
to prove anything here!
It is so new.

And how is that connect
to the discovery of the DNA ?


Moshek

[t)]

[Organic]

The discovery of the DNA is a beautiful example of a good science about our abilities to explore the power of simplicity in nature.

Please see: http://biologybooks.net/074321630X.html

[phoenixthoth]

i think i'm starting to understand your perspective. thank you; it is interesting.

[Russell E. Rierson]

Potential infinity is defined as a limit via Newton's calculus, while actual infinity is a Cantorian Cardinal number, which is a Platonic form, which is also a type of potential.

[abstract representation]--->[semantic mapping]--->[represented system]

[axiomatic]--->[Isomorphism]<---[Induction]

An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct; topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

A duality?

A point contains an infinite expanse of space and time?

Could it be, that the "absolute" infinity, is actually a dimensionless point?

[point]/[set of points] = point ?


0/N = 0


Since it is possible for a "computation" to be self aware, there must be platonic forms that are types of self aware algorithms:


The description of any entity inside the real universe can only be with reference to other things in the universe. Space is then relational, and the universe, self referential. For example, if an object has a momentum, that momentum can only be explained with respect to another object within the universe. Space then becomes an aspect of the relationships between things in reality. It becomes analogous to a sentence, and it is absurd to say that a sentence has no words in it. So the grammatical structure of each sentence[space] is defined by the relationships that hold between the words in it. For example, relationships like object-subject or adjective-noun. So there are many different grammatical structures composed of different arrangements of words, and the varied relationships between them.

Language describes the universe, because the universe is isomorphic to a description on some level, and reality can only refer to itself, because, there is nothing outside of ..."total existence" which becomes equivalent to a self referential system, which must be a self aware system. Since descriptions make distinctions, or references to other entities, and distinctions are tautologically logical, [A or ~A], reality is logical, in that its contents can be described by a language. The contents within reality are distinctive entities, individually different from the others, yet consisting of the same foundational substance.


[<-[->[<-[U]->]<-]->]

Universe = Zero



On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

Einstein's equation basically says

Einstein Tensor [G] = Stress-Energy Tensor [T]

[spacetime geometry] determines [matter-energy's path] = geodesic.

[Matter-energy] determines [spacetime geometry] = non-Euclidean geometry.

.
Conservation of momentum energy is explained as an automatic consequence of the zero boundary of a boundary. Where conservation of energy-momentum means no creation or destruction of energy momentum in a 4D region of spacetime [4D cube] The integral of "creation events" i.e. the integral of
d*T for energy momentum, over the 4D region is required to be zero, and gives the conservation of momentum energy. The mathematical machinery for identically meeting the conservation laws is the boundary of a boundary equals zero.

[spacetime tells mass]<===[geodesic path for particle]===>[mass tells spacetime]

Waves are ripples in a basic medium. Einstein explains that the ether is unecessary as a medium, so the ripples are vibrations of spacetime itself, if, mass-energy is a form of condensed space-time.

As the ripples intersect with each other, it becomes a domino effect with the ripples continually increasing in density. Very similar to taking a penny and doubling it as an iterative sequence.

2, 4, 8, 16, 32, 64, 128, 256, ... 2^n

Since the ripples are increasing in density they are "compressed" . As spacetime becomes compressed, matter is re-configured as a balancing effect, so the force of gravity and accelerations are perceived as they presently are.

[<->[<->[<->[U]<->]<->]<->]

The increasing spacetime density must be background independent.

Actually, spacetime does not really need to be "sliced up" in that it can proceed in discrete steps, yet, still be continuous.

[density 1]--->[density 2]--->[density 3]---> ... --->[density n]


A quote from the book "The Expanding Universe" by Sir Arthur Eddington:



All change is relative. The universe is expanding relatively to our common standards; our common standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom" .


Quantum mechanics leads us to the realization that all matter-energy can be explained in terms of "waves". In a confined region(i.e. a closed universe or a black hole) the waves exists as STANDING WAVES In a closed system, the entropy never decreases.

The analogy with black holes is an interesting one but if there is nothing outside the universe, then it cannot be radiating energy outside itself as black holes are explained to be. So the amount of information i.e. "quantum states" in the universe is increasing. We see it as entropy, but to an information processor with huge computational capabilities, it is compressible information.

Quantum field theory calculations where imaginary time is periodic, with period 1/T are equivalent to statistical mechanics calculations where the temperature is T. The periodic waveforms that are opposed yet "in phase" would be at standing wave resonance, giving the action.

Periodicity is a symmetry. Rotate into the complex plane and we have
real numbers on the horizonal axis and imaginary numbers on the
vertical axis. So a periodic function could exist with periodicity
along both the imaginary AND the real axis. Such functions would have
amazing symmetries. Functions that remain unchanged, when the complex
variable "z" is changed.

f(z)---->f(az+b/cz+d)

Where the elements a,b,c,d, are arranged as a matrix, forming an
algebraic group. An infinite number of possible variations that
commute with each other as the function f, is invariant under group
transformations. These functions are known as "automorphic forms".

Topologically speaking, the wormhole transformations must be
invariant with regards to time travel. In other words, by traveling
backwards in time, we "complete" the future, and no paradoxes are
created.

So when spacetime tears and a wormhole is created, it must obey
certain transformative rules, which probably appear to be
discontinuities from a "3-D" perspective, but really, these
transformations are continuous?


[v1+v2]/[1+ v1v2/c^2]

c+c = c

aleph_0 + aleph_0 = aleph_0

0 + 0 = 0

Gravity exists because the information density of space-time is increasing. This creates a "pressure force" where processed space, compresses mass-energy, and mass-energy reacts by compressing space. The process is "time", which becomes dilated due to the increased information density of massive objects.
Stephen Hawking's excellent book, "Universe in a Nutshell", explains holography as a phenomenon of interference of wave patterns. Light from a laser is split into two separate beams, one bounces off the object and gets reflected onto a photo-sensitized plate. The other beam is reflected into a lens and collides with the reflected light of the object. When a laser is shone through the developed plate, a fully three dimensional image of the original object is created.

According to conventional theories, the surface area of the horizon surrounding a black hole, measures its entropy, where entropy is defined as a measure of the number of internal states that the black hole can be in without looking different to an outside observer, who can only measure mass, rotation and charge. This leads to another theory which states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system. So the theorists came to realize that the information associated with all phenomena in the three dimensional world, can be stored on its two dimensional boundary, like a holographic image.

S' = S_m + A/4

Since entropy can also be defined as the number of states within a region of space, and the entropy of the universe must always increase, the next logical step is to realize that the spacetime density, i.e. the information encoded within a circumscribed region of space, must be increasing in the thermodynamic direction of time.

Entropy of thermodynamics and entropy of Shannon, are equivalent concepts, because the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information needed to implement any particular combination, or arrangement. The two entropies also appear to have differences, superficially. Thermodynamic entropy interpreted in units of energy divided by temperature, while, the Shannon entropy is interpreted in terms of bits, being essentially dimensionless. The difference is a matter of convention.

[Organic]

Hi Russell E. Rierson,

This is a very interesting post.

Let me ask a question:

What if space-time is the dynamic results of opposite things?

In this case any model that based on integer dimensions is too trivial.

In my opinion, space-time model which is based on this point of view is less trivial:

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

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

http://www.geocities.com/complementarytheory/CL-CH.pdf

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

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

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

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

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

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

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



Please reply your remarks.

Thank you.

Yours,

Organic

[Organic]

Dear Phoenixthoth,

Once some PF mentor answerd (after i asked for some help):

This may help. Its his first post in his other thread.

-----------------------------------------------------------------
In the attached address you can find A new approach for the definition of a NUMBER...
-----------------------------------------------------------------

Today I went to the bank and tried to define a new definition of "exchange rate" but for some reason they weren't buying. Hmm, I don't understand why not...

Your comment was:

(PF mentor's name), no, it doesn't help. your comments remind me of those who didn't like the idea of irrational, transcendental, hyperreal, or complex numbers. gauss, as far as in know, invented a new kind of number and they were used in his PhD thesis. ...


I am glad that I can give you back something that you find as interesting.

Yours,

Organic

[Russell E. Rierson]

Originally posted by Organic
Hi Russell E. Rierson,

This is a very interesting post.

Let me ask a question:

What if space-time is the dynamic results of opposite things?

Thank you for the links Organic, I will be reading them for awhile.

A point without another "reference" does not exist; the opposite of a thing distinguishes it from the thing itself. What is the dynamic of space-time? Is it a ratio?

When space is taken as a measure of length, space/time is the speed of light in vacuum for a photon of light:

space/time = c


Where, length = perception of separation between two reference points.

E = mc^2

E/momentum = E/p = c

energy/momentum = space/time

What is the EPR "superluminal?" connection? A shortcut through configuration space? Phase space?

A point can be defined as an "infinitesimal". The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system.

[phoenixthoth]

organic,
i remember that as well. of course if all you want to do is work with money, all you need is rational numbers. if you want to work with higher physics, you need the complex numbers gauss helped introduce. i don't know what conway's surreal numbers are good for but they are interesting and isn't that enough? hyperreal numbers can actually be used to prove statements in real analysis, though every analyst i've spoken with says they'd rather use real analysis. i will always encourage the search for something new even though i think that the search may just be a frustrating effort of spinning one's wheels and not doing anything essentially new. but why discourage the attempt? often when one does something like that, it gives them better understanding of the rules they are bending, of regular numbers.

phoenix

[Organic]

Hi Russell E. Rierson,


Pure Math does not use 'time or 'process' as a part of its system, so if we are talking about the dimensions of a macro system like the universe (where the word 'dynamic' has a meaning) I think we can use the combination "dynamic results" .

In this case I am talking about using another way to look on dimensions, which is not static and it is not based on integers.

[moshek]

You are an amassing thinker, Russell E. Rierson.
And my first opinion is Thank you.

Moshek
[:))]

[Organic]

Dear phoenixthoth,

We will never know if we don't go.

[moshek]

It is the time to go
for Hilbert vision.

Moshek

----------------------------------------------------------------

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!

D.Hilbert 1900

[moshek]

Dear Organic,

It was proved by Franklin 1934 that you need 6 color in general to color a map in Klein bottle. The geometry shape in 4 dimension of Non-Euclidian mathematics . That represent the duality ( symbol, action) of any concept in mathematics a it appear in numbers. they are meet in the two size of the bottle as in your complementary theory. by positive interpretation to Godel theorem (1931) and by this present a solution to Hilbert 6 problem (1900 Paris).

your's

Moshek
[:))]