Boolean Logic cannot deal with infinitely many objects

[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]

Cantor prove is fine

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]

nice

That is a very good question to Organic.

So let's wait to his answer!

 

[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 afraid

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

Does Organic
is really a monkey
like he is looking ?

 

[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 really like the way you think
So i just want to triger you.

well i don't have any picture
like that you have,
but i want to tend to infinity
even more than Alef0

So really 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]

What are you trying to solved?

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]

New Millenium for mathematics

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]

i answer to you in the wrong place

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 don't 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.