Boolean Logic cannot deal with infinitely many objects

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The discussion centers on the limitations of Boolean logic and Cantor's diagonalization method when applied to infinite sets. The author argues that Cantor's method cannot account for all combinations in an infinite list, leading to the conclusion that 2^aleph0 equals aleph0, which contradicts established mathematical principles. They present examples of finite combinations and assert that the diagonalization results do not yield new combinations, implying that Boolean logic fails with infinitely many objects. The conversation includes critiques of the author's reasoning and calls for a better understanding of mathematical concepts, particularly Cantor's argument. Ultimately, the thread highlights the complexities and misunderstandings surrounding infinite sets and their treatment in mathematics.
  • #31
(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.
 
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  • #32
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]
 
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  • #33
will you be satisfy if...

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


:smile: :smile:
 
  • #34
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
 
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  • #35
On Thusday

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)]
 
  • #36
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.
 
  • #37
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?
 
  • #38
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
 
  • #39
|N| is a fixed constant. How does it approach something?
 
  • #40
nice

That is a very good question to Organic.

So let's wait to his answer!
 
  • #41
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
 
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  • #42
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?
 
  • #43
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
 
  • #44
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 ?
 
  • #45
just because we can't actually count to infinity doesn't mean infinity doesn't exist...

moshek: that's not very nice of you
 
  • #46
Hi Moshek,

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



Orgainc
 
  • #47
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.
 
  • #48
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
 
  • #49
Hi Moshek,


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


Yours,


Organic
 
  • #50
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?

:smile:
 
  • #51
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.
 
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  • #52
|\mathbb{N}| is something. *shrug*
 
  • #53
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...
 
  • #54
|N| is the cardinal of infinitely many objects that approaching aleph0, where aleph0=actual infinity.
 
  • #55
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.
 
  • #56
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 :wink:
 
  • #57
can you explain it to me?
sounds like a question clinton would ask. not as simple as one might think, actually.
 
  • #58
i answer to you in the wrong place

Sory Phoenixthoth
Moshek
 
  • #59
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.
 
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  • #60
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.
 

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