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.
  • #61
Hi HallsofIvy,



Please show us how you can use the content of {}(=emptiness) or the content of {__}(=fullness) as an input, by Math language.
 
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  • #62
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.
 
  • #63
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
 
  • #64
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.
 
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  • #65
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:

=?
_=?
 
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  • #66
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.
 
  • #68
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
 
  • #69
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
 
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  • #70
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.
 
  • #71
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?
 
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  • #72
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.
 
  • #73
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
 
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  • #74
But its sooooo complicated Organic. Can you at least tell me what you are trying to prove?
 
  • #75
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
 
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  • #76
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.
 
  • #78
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.
 
  • #79
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
 
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  • #80
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...:smile:
 
  • #81
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
 
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  • #82
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 referring 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?
 
  • #83
{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.
 
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  • #84
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  • #85
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
 
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  • #86
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.
 
  • #87
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

Organic I'm still waiting for your response on which of the sets are actual and which potential infinities...
 
  • #88
Is there any x such that x \in \{\_\_\}?
 
  • #90
no...more...pdfs...please...
 

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