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Old 27th July 2020, 06:19 AM   #121
doronshadmi
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Now here is my question to you:

Please complete the following definition:

|N| > any given n iff ...
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 27th July 2020, 06:53 AM   #122
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Originally Posted by doronshadmi View Post
EDITED:

The bus stands at some station (no matter what station) means D is finite.

The bus does not stand at any station means D is non-finite.

That's all we need in order to distinguish between the finite and the non-finite.

No decision of involved in order to distinguish between stands or does not stand.

You have provided no means to determine what the bus will do for a given D.

You have therefore provided not means to distinguish finite from non-finite.
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Old 27th July 2020, 06:56 AM   #123
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Originally Posted by doronshadmi View Post
Now here is my question to you:

Please complete the following definition:

|N| > any given n iff ...

Why would I do that? That has no bearing on anything I have offered.
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Old 28th July 2020, 12:06 AM   #124
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Originally Posted by jsfisher View Post
You have provided no means to determine what the bus will do for a given D.

You have therefore provided not means to distinguish finite from non-finite.
D is a placeholder for sets, therefore D is not any particular set, so I do not have to provide means to determine what the bus will do for a given D.

If the bus stands (|D| is determined by some n) then D is finite.

If the bus does not stand (|D| is not determined by some n) then D is non-finite.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 28th July 2020 at 12:18 AM.
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Old 28th July 2020, 12:17 AM   #125
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Originally Posted by jsfisher View Post
Why would I do that? That has no bearing on anything I have offered.
Because by traditional mathematics |N| > any given n

Since you represent here the reasoning of traditional mathematicians about the discussed subject (where by traditional mathematics |N| > any given n), you are able to complete the following definition:

|N| > any given n iff ...
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 28th July 2020, 04:10 AM   #126
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Originally Posted by doronshadmi View Post
Because by traditional mathematics |N| > any given n
I have told you my approach to defining cardinality and what it means for a set to be non-finite. What you wrote never came up.
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Old 28th July 2020, 04:13 AM   #127
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Originally Posted by doronshadmi View Post
D is a placeholder for sets, therefore D is not any particular set, so I do not have to provide means to determine what the bus will do for a given D.

If the bus stands (|D| is determined by some n) then D is finite.

If the bus does not stand (|D| is not determined by some n) then D is non-finite.
You have in effect told us you define finiteness in terms of cardinality. Therefore, you need to define cardinality. Part of that will be how to determine whether |D| is some member of N, or not.
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Old 28th July 2020, 05:09 AM   #128
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Originally Posted by jsfisher View Post
You have in effect told us you define finiteness in terms of cardinality. Therefore, you need to define cardinality. Part of that will be how to determine whether |D| is some member of N, or not.
Let D be a placeholder for any set.

Let N be the set of natural numbers.

Definition 1: Cardinality is the "size" of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to" means that Cardinality is not necessarily some particular N member, yet it is at the domain of N, where N does not have the greater successor.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 28th July 2020, 06:28 AM   #129
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I have a set D which is the set of combinations of the English vowels (a, e, i, o, u). How do I determine the set's cardinality?

Last edited by Little 10 Toes; 28th July 2020 at 06:29 AM. Reason: Formatting
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Old 28th July 2020, 06:58 AM   #130
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Originally Posted by Little 10 Toes View Post
I have a set D which is the set of combinations of the English vowels (a, e, i, o, u). How do I determine the set's cardinality?
By the number of the combinations that can be done by 5 distinct things (the English vowels, in this case).

Please see https://en.wikipedia.org/wiki/Combination
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 28th July 2020, 07:07 AM   #131
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The element aa is an acceptable combination.

How do you determine the set's cardinality?

Last edited by Little 10 Toes; 28th July 2020 at 09:00 AM. Reason: Bolded the element aa; corrected the question
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Old 28th July 2020, 09:11 AM   #132
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Originally Posted by Little 10 Toes View Post
The element aa is an acceptable combination.

How do you determine the set's cardinality?
If aa is an element (https://en.wikipedia.org/wiki/Element_(mathematics)) of D, than |D| is at least 1.

Generally, you do not provide enough information about your set (which its members are combinations) since you did not determine k (see https://en.wikipedia.org/wiki/Combination).
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 28th July 2020 at 09:32 AM.
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Old 28th July 2020, 09:22 AM   #133
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Originally Posted by doronshadmi View Post
Let D be a placeholder for any set.

Let N be the set of natural numbers.

Definition 1: Cardinality is the "size" of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to" means that Cardinality is not necessarily some particular N member, yet it is at the domain of N, where N does not have the greater successor.
That didn't cut it as a definition that last time you posted it. Reposting it now doesn't improve on that.

The definition body itself is "grammatical" nonsense. You have presented "Cardinality is the 'size' of D" as a true/false proposition. So, in some cases cardinality is the "size" of D, and in other cases it is not. Great, I suppose, but what does cardinality actually mean?

As for "'related to' means that Cardinality is not necessarily some particular N member", you haven't described with any specificity what 'related to' means.

From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of some set, S?

From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of a set, {a,b,c,d}, to be 4 and not be 21?
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Last edited by jsfisher; 28th July 2020 at 11:04 AM.
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Old 28th July 2020, 09:40 AM   #134
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Originally Posted by doronshadmi View Post
If aa is an element (https://en.wikipedia.org/wiki/Element_(mathematics)) of D, than |D| is at least 1.

Generally, you do not provide enough information about your set (which its members are combinations) since you did not determine k (see https://en.wikipedia.org/wiki/Combination).
Why do I care about k?

You did not answer the question. In my example, how do you determine the cardinality of set D?
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Old 28th July 2020, 09:04 PM   #135
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Please note doronshadmi, I asked the same question three times. Yes, there are ways to determine combinations. Your answer of, "then |D| is at least 1" is nonsense. Why? Because that was not what I asked for. I will ask it again very clearly.

In my example, how do you determine the cardinality of set D?
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Old 28th July 2020, 11:48 PM   #136
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Originally Posted by jsfisher View Post
You have presented "Cardinality is the 'size' of D" as a true/false proposition. So, in some cases cardinality is the "size" of D, and in other cases it is not.
Let N be the set of natural numbers.

Cardinality is the 'size' of D, whether it is determined by a particular N member (in this case |D|=particular n and D is called finite set) or it is determined according the fact that N does not have the greatest successor (in this case |D| is not any particular n, yet we stay at N domain and D is called non-finite set).

So being at N domain this is the true state of |D|, whether it is determined by some particular n, or not.

Code:
 
p="Cardinality is the 'size' of D" 

q="|D| is at N domain"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T

Definition 1: Cardinality is the 'size' of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to the members of N" simply means that |D| is at N domain, whether it is determined by some particular n, or not (in this case the fact that N does not have the greatest successor is taken).

Originally Posted by jsfisher View Post
From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of a set, {a,b,c,d}, to be 4 and not be 21?

The definitions in http://www.internationalskeptics.com...7&postcount=98 have nothing to do with defining specific values of cardinality, so your question simply show that you basically does not follow after what they are actually define.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 29th July 2020 at 12:30 AM.
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Old 29th July 2020, 04:14 AM   #137
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Originally Posted by doronshadmi View Post
The definitions in http://www.internationalskeptics.com...7&postcount=98 have nothing to do with defining specific values of cardinality, so your question simply show that you basically does not follow after what they are actually define.
So, you agree, your "definition" of cardinality is insufficient to determine your concept of cardinality.
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Old 29th July 2020, 06:07 AM   #138
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Originally Posted by jsfisher View Post
So, you agree, your "definition" of cardinality is insufficient to determine your concept of cardinality.
No jsfisher, my definition of cardinality is sufficient to determine the concept of cardinality, which is not limited to your restriction to define some particular cardinality by using the definition.

It is called generalization, which enable to use it in order to distinguish between finite and non-finite sets, without the need for special axioms (which in case of ZF axiom of infinity and the axiom schema of restricted comprehension, they do not provide a non-finite set that its cardinality > any given N member, but such a set is arbitrarily determined. Also the very notion of axiom schema as a non-finite set of axioms, is arbitrarily determined).
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 29th July 2020 at 06:08 AM.
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Old 29th July 2020, 09:15 AM   #139
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The best that the ZF axiom of infinity and the axiom schema of restricted comprehension (with more ZF axioms) establish, is the inductive set known by the name "von Neumann ordinals" that does not have the greatest successor, so the term all in the following equate is arbitrarily plugged in:

Originally Posted by jsfisher View Post
The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...).

A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element.

So, a set X is a non-finite set if and only if |N| <= |X|, where N is the minimal set established by the Axiom of Infinity.
Now let's look at this:
Originally Posted by jsfisher View Post
Since |N| <= |N|, done!
This is a concrete example of an arbitrary expression.

----------

One may ask if |N| > the cardinality of any given N member (where in our case, the members of N are von Neumann ordinals, as constructed by based on the two properties of ZF axiom of infinity).

Since the cardinality of any given member of N is finite and the term all is arbitrarily plugged in (bye bye to the fact that N does not have the greatest successor, which actually prevents the term all) |N| > cardinality of any given N member, is inevitable by ZF.

Please look at your reply in http://www.internationalskeptics.com...&postcount=123 and the following quote:
Originally Posted by jsfisher View Post
If you want to label the members of the von Neumann ordinals with natural numbers, go right ahead, but that's your doing, not a requirement of mine.
The cardinality of any given N member (where N is actually the set of von Neumann ordinals) can't be but some natural number, so "not a requirement of mine" has no basis whatsoever.

Therefore please complete the following definition (by using ZF reasoning):

|N| > any given n iff ...
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 29th July 2020 at 09:50 AM.
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Old 29th July 2020, 09:47 AM   #140
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Originally Posted by doronshadmi View Post
No jsfisher, my definition of cardinality is sufficient to determine the concept of cardinality, which is not limited to your restriction to define some particular cardinality by using the definition.
Then you haven't defined cardinality at all. You have only told us about some of the characteristics you think it must have.
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Old 29th July 2020, 09:59 AM   #141
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Originally Posted by jsfisher View Post
From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of a set, {a,b,c,d}, to be 4 and not be 21?
By my definitions |{a,b,c,d}| can't be but the member of N called 4, even if they are not directly used in order to define some particular cardinality.

Cardinality is directly defined by my definition, but not some particular one.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 29th July 2020 at 10:12 AM.
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Old 29th July 2020, 10:03 AM   #142
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Originally Posted by jsfisher View Post
Then you haven't defined cardinality at all. You have only told us about some of the characteristics you think it must have.
No, it is based on iff (your requirement) as seen in http://www.internationalskeptics.com...&postcount=136.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 29th July 2020, 02:04 PM   #143
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Originally Posted by doronshadmi View Post
The best that the ZF axiom of infinity and the axiom schema of restricted comprehension (with more ZF axioms) establish, is the inductive set known by the name "von Neumann ordinals" that does not have the greatest successor...
That's the best? You seem to have overlooked just how foundational set theory is to the entire body of Mathematics.

Quote:
...so the term all in the following equate is arbitrarily plugged in....
Trying to quibble over the meaning of 'all' is just silly. You, who has repeatedly failed to define any of the terms you use in so many fantastical ways, are not qualified to write on this matter.

Stop with the countless diversions, please. You've made claims. You need to defend your claims, and not jump to any topic you can that isn't among your claims.

Focus: Start with your definition of cardinality as you are using the term.
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Old 29th July 2020, 02:21 PM   #144
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Originally Posted by doronshadmi View Post
No, it is based on iff (your requirement) as seen in http://www.internationalskeptics.com...&postcount=136.
I have no such requirement.

If there is something to be defined that is boolean valued, say, for example, whether a set is non-finite, then an if-and-only-if construct is perfectly reasonable. A set is non-finite if and only if something else (that is already well-defined) is true.

But you aren't expressing things as boolean valued. An if-and-only-if construction makes absolutely no sense for that, except perhaps to someone who really doesn't understand basic mathematical logic.

Moreover, you want the cardinality of some sets to be whole numbers, but the gibberish you presented doesn't even begin to define cardinality in a way that the cardinality of a set can be determined.
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Old 29th July 2020, 02:23 PM   #145
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Originally Posted by doronshadmi View Post
By my definitions |{a,b,c,d}| can't be but the member of N called 4, even if they are not directly used in order to define some particular cardinality.

Cardinality is directly defined by my definition, but not some particular one.
Nothing in your posted "definitions" lead anyone to conclude the cardinality of {a,b,c,d} is 4.

Prove me wrong. Show your work.
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Old 30th July 2020, 02:43 AM   #146
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Originally Posted by jsfisher View Post
Trying to quibble over the meaning of 'all' is just silly.
No, it is at the heart of our discussion.

By trying to force "all" on a set that the existence of its greatest successor is not satisfied, you actually eliminate the very existence of an endless collection (where a set is some particular case of collection).

For example, please define the greatest successor of the set of von Neumann ordinals.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 30th July 2020 at 02:49 AM.
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Old 30th July 2020, 04:05 AM   #147
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Originally Posted by jsfisher View Post
Nothing in your posted "definitions" lead anyone to conclude the cardinality of {a,b,c,d} is 4.

Prove me wrong. Show your work.
Very simple:

Definition 1: Cardinality is the 'size' of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to the members of N" simply means that |D| is at N domain, whether it is determined by some particular n, or not (in this case the fact that N does not have the greatest successor is taken).

D={a,b,c,d}

|D|=|{a,b,c,d}| (the 'size' of D)

By constricting the natural numbers in terms of sets, as done by von Neumann |{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}| = 4

Since {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}} and {a,b,c,d} are bijective, |D| can't be but 4.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 30th July 2020, 04:12 AM   #148
jsfisher
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Originally Posted by doronshadmi View Post
Since {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}} and {a,b,c,d} are bijective, |D| can't be but 4.

Nothing in your so-called definition supports this leap. Perhaps you'd like to include the word 'bijective' in what you've written, at the very least.

ETA:
Quote:
By constricting the natural numbers in terms of sets, as done by von Neumann |{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}| = 4
I think you meant that {{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}} = 4 (without the vertical bars).
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Last edited by jsfisher; 30th July 2020 at 04:22 AM.
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Old 30th July 2020, 04:43 AM   #149
doronshadmi
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Originally Posted by jsfisher View Post
Nothing in your so-called definition supports this leap. Perhaps you'd like to include the word 'bijective' in what you've written, at the very least.
There is no leap.

Definition 2: The cardinality between the two sets A and B is a relative measure of their "sizes", where ( ((|A| < |B|) iff (|B| > |A|)) OR ((|A| = |B|) iff (|B| = |A|)) ).

The highlighted part is bijection.

As for von Neumann construction, natural numbers and Cardinality:

Code:
{
   -------------->
0   | 	= |∅|
1   | 	= |{∅}|
2   | 	= |{ ∅, {∅} }|
3   | 	= |{ ∅, {∅} , {∅, {∅}} }|
4   | 	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... V

}
The horizontal direction is any finite cardinality.

The vertical downward direction is non-finite cardinality (the existence of the greatest successor is not satisfied).

In both cases you do not leave von Neumann domain.
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 30th July 2020 at 04:50 AM.
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Old 30th July 2020, 04:55 AM   #150
jsfisher
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Originally Posted by doronshadmi View Post
There is no leap.

Definition 2: The cardinality between the two sets A and B is a relative measure of their "sizes", where ( ((|A| < |B|) iff (|B| > |A|)) OR ((|A| = |B|) iff (|B| = |A|)) ).

The highlighted part is bijection.
You might want to double check your understanding of what a bijection is. That which you highlighted is a tautology, but it is nothing like a bijection.
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Old 30th July 2020, 06:07 AM   #151
doronshadmi
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Originally Posted by jsfisher View Post
You might want to double check your understanding of what a bijection is. That which you highlighted is a tautology, but it is nothing like a bijection.
EDIT:
Eeach element of A is mapped to exactly to one element of B and each element of B is mapped to exactly to one element of A , so in terms of cardinality |A|=|B| iff |B|=|A|.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 30th July 2020 at 06:44 AM.
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Old 30th July 2020, 06:11 AM   #152
doronshadmi
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Originally Posted by jsfisher View Post
I think you meant that {{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}} = 4 (without the vertical bars).
In case of cardinality |{{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}}| = 4
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 30th July 2020, 06:48 AM   #153
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Originally Posted by doronshadmi View Post
In case of cardinality |{{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}}| = 4

Nowhere in your "definitions" does this factoid appear.
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Old 30th July 2020, 08:10 AM   #154
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Originally Posted by doronshadmi View Post
EDIT:
Eeach element of A is mapped to exactly to one element of B and each element of B is mapped to exactly to one element of A , so in terms of cardinality |A|=|B| iff |B|=|A|.
Let's try that out:

Let A = {a, b, c}, and let B = {}.

|A| = |B| ? Well, impossible to tell from Doronshadmi's "definitions", but I will assume Doronshadmi would say "false" since he wants |A| to be 3 and |B| to be 0. (We'd reach the same final result were the answer "true".)

|B| = |A| ? "False".

Therefore: |A| = |B| iff |B| = |A| is "true".

Conclusion: Each element of A is mapped to exactly to one element of B and each element of B is mapped to exactly to one element of A.
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Old 31st July 2020, 12:29 AM   #155
doronshadmi
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Originally Posted by jsfisher View Post
Let's try that out:

Let A = {a, b, c}, and let B = {}.

|A| = |B| ? Well, impossible to tell from Doronshadmi's "definitions", but I will assume Doronshadmi would say "false" since he wants |A| to be 3 and |B| to be 0. (We'd reach the same final result were the answer "true".)

|B| = |A| ? "False".

Therefore: |A| = |B| iff |B| = |A| is "true".

Conclusion: Each element of A is mapped to exactly to one element of B and each element of B is mapped to exactly to one element of A.
Thank you, you are right.

"Each element of A is mapped to exactly to one element of B" is unclear since, for example, it can be also surjection, and so is the case about the other direction.

So, back to work:

D is a placeholder for any given set.

Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.

Code:
 
p="Cardinality is the 'size' of D" 

q="|D| is at von Neumann domain"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T
---------------------------------------------------------------------
As about von Neumann domain, natural numbers and Cardinality:

Code:
{
   -------------->
0   | 	= |∅|
1   | 	= |{∅}|
2   | 	= |{ ∅, {∅} }|
3   | 	= |{ ∅, {∅} , {∅, {∅}} }|
4   | 	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... V

}
(The horizontal direction is any finite cardinality.

The vertical downward direction is non-finite cardinality (the existence of the greatest successor is not satisfied).

In both cases one does not leave von Neumann domain.)
---------------------------------------------------------------------

Definition 2: N is the set of finite cardinalities iff any given nN member is determined by a particular von Neumann member ( as follows: )

Code:
{
   -------------->
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... 

}
Definition 3: D is called finite iff it is bijective with some particular von Neumann member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member (the fact that von Neumann set does not have the greatest successor, is taken), yet we do not leave N domain.
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 31st July 2020 at 01:07 AM.
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Old 31st July 2020, 03:52 AM   #156
doronshadmi
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Definition 2 correction:

Definition 2: N is the set of finite cardinalities iff each given nN is determined by each von Neumann member ( as follows: )

Code:
{
   -------------->
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... 

}
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 31st July 2020 at 04:08 AM.
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Old 31st July 2020, 06:55 AM   #157
doronshadmi
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Definition 2 correction:

Definition 2: N is the set of finite cardinalities iff each given nN is determined by its corresponding von Neumann member ( as follows: )

Code:
{
   -------------->
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... 

}
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 31st July 2020, 08:44 AM   #158
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Doronshadmi, please start using different set letters. N is usually used for the set of natural numbers.
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Old 31st July 2020, 04:38 PM   #159
jsfisher
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Originally Posted by doronshadmi View Post
Thank you, you are right.
Yes.

Quote:
"Each element of A is mapped to exactly to one element of B" is unclear since, for example, it can be also surjection, and so is the case about the other direction.
No.

The whole point of my post was that you are misusing the "if and only if" construct. Whatever it is you think what you write might mean, you are making it gibberish by what you do with "iff".

There is another thing you need to do, too: Your "definitions" must not have forward references. Your definitions may not rely on any terms that aren't yet defined. If Definition 1 depends on Definition 4, for example, then you need to reorder your definitions to move Definition 1 to a new spot after Definition 4.
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Last edited by jsfisher; 31st July 2020 at 04:42 PM.
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Old 31st July 2020, 10:08 PM   #160
doronshadmi
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Ok here is my corrected version.

D is a placeholder for any given set.

Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.

Code:
 
p="Cardinality is the 'size' of D" 

q="|D| is at von Neumann domain"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T
---------------------------------------------------------------------
As about von Neumann domain, natural numbers and Cardinality:

Code:
{
   -------------->
0   | 	= |∅|
1   | 	= |{∅}|
2   | 	= |{ ∅, {∅} }|
3   | 	= |{ ∅, {∅} , {∅, {∅}} }|
4   | 	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... V

}
(The horizontal direction is any finite cardinality.

The vertical downward direction is non-finite cardinality (the existence of the greatest successor is not satisfied).

In both cases one does not leave von Neumann domain.)
---------------------------------------------------------------------

Definition 2: N is the set of finite cardinalities iff each given nN is determined by its corresponding von Neumann member ( as follows: )

Code:
{
   -------------->
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... 

}
Definition 3: D is called finite iff it is bijective with some particular von Neumann member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member (the fact that von Neumann set does not have the greatest successor, is taken), yet we do not leave N domain.
__________________
That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 31st July 2020 at 10:09 PM.
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