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Old 1st August 2020, 10:57 AM   #161
jsfisher
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Originally Posted by doronshadmi View Post
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.

You continue to abuse that poor if-and-only-if relation.

What is "von Neumann domain"? What does it mean to be "at" it?

(No forward references, remember.)
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Old 4th August 2020, 03:16 AM   #162
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Originally Posted by jsfisher View Post
You continue to abuse that poor if-and-only-if relation.

What is "von Neumann domain"? What does it mean to be "at" it?

(No forward references, remember.)
In that case, please take what I wrote immediately after definition 1 (seen in http://www.internationalskeptics.com...&postcount=160) an put it before it.
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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 4th August 2020, 03:37 AM   #163
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There are three other definitions. Please arrange them in the order that you wish. No forward references please.
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Old 4th August 2020, 04:10 AM   #164
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Originally Posted by doronshadmi View Post
In that case, please take what I wrote immediately after definition 1 (seen in http://www.internationalskeptics.com...&postcount=160) an put it before it.
(1) That would remedy none of the three defects I cited.
(2) Your water; you carry it.
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Old 5th August 2020, 06:58 AM   #165
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Originally Posted by Little 10 Toes View Post
There are three other definitions. Please arrange them in the order that you wish. No forward references please.
They are ordered, such that definition 2 is based on definition 1, definition 3 is based on definition 2 and definition 4 is based on definitions 1 and 2.
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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 5th August 2020, 07:46 AM   #166
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Originally Posted by jsfisher View Post
(1) That would remedy none of the three defects I cited.
"von Neumann domain" is the 'horizontal' and 'vertical' directions of the following inductive set:

Code:
{
   -------------->
    | 	   ∅, 
    | 	   {∅}, 
    | 	   { ∅, {∅} }, 
    | 	   { ∅, {∅} , {∅, {∅}} }, 
    | 	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, 
... V

}
In both cases one is at von Neumann domain.

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
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 one is at N domain, which is equivalent to von Neumann domain).
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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; 5th August 2020 at 07:52 AM.
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Old 5th August 2020, 09:38 AM   #167
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Originally Posted by doronshadmi View Post
"von Neumann domain" is the 'horizontal' and 'vertical' directions of the following inductive set...
Sets don't have directions. What do you mean by "von Neumann domain"? Vague pictures won't help you define it.

Quote:
In both cases one is at von Neumann domain.
What does "at" mean in "one is at von Neumann domain"?

Quote:
D is a placeholder for any given set.

Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.
What does "'size' of D" mean? How does one determine whether |D| is "at von Neumann domain" without knowing what the cardinality of D is?
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Old 9th August 2020, 05:46 AM   #168
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Originally Posted by jsfisher View Post
Sets don't have directions.
I do not mean direction in terms of space, but in terms of guidance of notions.

von Neumann inductive set is not just a collection of finite sets (what I call "the 'horizontal' direction" in my diagram) but also the fact that the existence of the biggest set of this collection is not satisfied (what I call "the 'vertical downward' direction" in my diagram).

Originally Posted by jsfisher View Post
What do you mean by "von Neumann domain"? Vague pictures won't help you define it.

What does "at" mean in "one is at von Neumann domain"?
Both directions are called "von Neumann domain", where being "at" means that one takes both directions of von Neumann set and not only the fact that each member is a finite set.

Originally Posted by jsfisher View Post
What does "'size' of D" mean? How does one determine whether |D| is "at von Neumann domain" without knowing what the cardinality of D is?
Please look at the rest of my definitions in http://www.internationalskeptics.com...&postcount=166 .
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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; 9th August 2020 at 06:05 AM.
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Old 9th August 2020, 11:52 AM   #169
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Originally Posted by doronshadmi View Post
I do not mean direction in terms of space, but in terms of guidance of notions.
Doesn't matter what type of direction you meant, sets do not have direction.

Quote:
von Neumann inductive set
What set is that?

Quote:
...is not just a collection of finite sets (what I call "the 'horizontal' direction" in my diagram)
If you mean "collection of finite sets", then just say that. This continual interjection of meaningless terms as substitutes for less meaningless terms doesn't advance you point at all.

Quote:
...but also the fact that the existence of the biggest set of this collection is not satisfied (what I call "the 'vertical downward' direction" in my diagram).
What does "biggest set" mean? In what what does this "is not satisfied" characteristic have any bearing on anything?

Quote:
Both directions
Sets still don't have directions.

Quote:
...
And so on.
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Old 10th August 2020, 05:51 AM   #170
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Originally Posted by jsfisher View Post
If you mean "collection of finite sets", then just say that.
Ok.

D is a placeholder for any given set.

V is the von Neumann set of ordinals.

Definition 1: Cardinality is the 'size' of D iff |D| is determined by V.

Definition 2: N is the set of finite cardinalities iff each given nN is determined by its corresponding V member.

For example:
Code:
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
...
Definition 3: D is called finite iff it is bijective with some particular V member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member since V does not have the greatest 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.

Last edited by doronshadmi; 10th August 2020 at 05:52 AM.
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Old 10th August 2020, 06:29 AM   #171
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Originally Posted by doronshadmi View Post
Ok.

D is a placeholder for any given set.

V is the von Neumann set of ordinals.
Which set would that be? Seems unlikely you would be referring to the ordinals recognized by von Neumann since there are considerably more of those than you'd like to admit.

Quote:
Definition 1: Cardinality is the 'size' of D iff |D| is determined by V.
You still have that bogus "iff" in there. What does "is determined by" mean?
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Old 10th August 2020, 07:42 AM   #172
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Originally Posted by jsfisher View Post
... since there are considerably more of those than you'd like to admit.
Well, you can't admit (as seen in http://www.internationalskeptics.com...&postcount=139) that ZF did not establish ω or |N| exactly because of the bogus "all" that was arbitrarily added to it.

"all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied.

Originally Posted by jsfisher View Post
You still have that bogus "iff" in there. What does "is determined by" mean?
Moreover, |N| does not exist exactly because each member of N is determined by (easily understood term) its corresponding V member (the existence of the greatest successor is not satisfied).

The highlighted part is bogus since ZF did not define N as a non-finite set (in the actual sense).
Originally Posted by jsfisher View Post
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.
The minimal set established by the Axiom of Infinity is V, such that any given N member is the cardinality of its corresponding V member.

This is exactly the reason why you can't 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.

Last edited by doronshadmi; 10th August 2020 at 07:44 AM.
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Old 10th August 2020, 08:54 AM   #173
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doronshadmi, please fix your third message link. It does not go to the message with the quote.
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Old 10th August 2020, 10:31 AM   #174
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Originally Posted by doronshadmi View Post
Well, you can't admit (as seen in http://www.internationalskeptics.com...&postcount=139) that ZF did not establish ω or |N| exactly because of the bogus "all" that was arbitrarily added to it.
Your continual attempts to disprove definitions are pointless. Axiomatic set theories allow for infinite sets, and those sets contain "all" of there members.

Quote:
"all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied.
You still haven't defined this set, V. You haven't yet defined what "greatest successor" means. You haven't shown any relevance to set theory.
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Old 11th August 2020, 04:19 AM   #175
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Originally Posted by jsfisher View Post
Your continual attempts to disprove definitions are pointless.
Look jsfisher, by this categorical determination, you put the notion of definition as some kind of religious dogma.

It is not hard to understand that it is pointless to discuss with a given person about his\her dogmas.

Originally Posted by jsfisher View Post
Axiomatic set theories allow for infinite sets, and those sets contain "all" of there members.
You did not show all along our discussion that, for example, ZF consistently and rigorously establishes an infinite set that contains all of its members.

In order to be more concrete, please show exactly how ZF establishes ω as a limit ordinal of all smaller ordinals (https://en.wikipedia.org/wiki/Ordina...limit_ordinals):
Quote:
Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is α ∪ { α } since its elements are those of α and α itself.[3]

A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is the limit in a topological sense of all smaller ordinals (under the order topology).
Time after time the term "all" is used "out of the blue" without any reasoning behind it.

So, please demonstrate ZF reasoning, which actually consistently and rigorously establishes an infinite set that contains all of its members (for example: by ZF reasoning {0,1,2,3,4,5,...} is an infinite set that contains all of its members in spite of the fact that there is no largest natural number.

Quote:
Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:

There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.

So in the following sequence:

0, 1, 2, …, ω, ω+1

ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
No matter what symbolic maneuvers are used, the fact that the, so called, infinite {0,1,2,3,4,5,...} does not have the largest member (the terms "all" does not hold) actually prevents the existence of ω as a limit ordinal after 0, 1, 2, …

Since ω is not established, the very notion of the difference between cardinal numbers and ordinal numbers, in case of infinite sets, has no basis whatsoever.
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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; 11th August 2020 at 04:42 AM.
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Old 11th August 2020, 05:54 AM   #176
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Originally Posted by doronshadmi View Post
Look jsfisher, by this categorical determination, you put the notion of definition as some kind of religious dogma.

It is not hard to understand that it is pointless to discuss with a given person about his\her dogmas.
Interesting bit of projection, there.

Quote:
You did not show all along our discussion that, for example, ZF consistently and rigorously establishes an infinite set that contains all of its members.
Well, that would get back to defining what is meant by "an infinite set", which you have failed to express. My only claim related to this is that the Axiom of Infinity establishes the existence of a certain non-empty set. Of necessity, set's existence guarantees it has all of its members. That goes along with the definition of "all", but since you feel free to disregard meaning....

Quote:
In order to be more concrete, please show exactly how ZF establishes ω as a limit ordinal of all smaller ordinals (https://en.wikipedia.org/wiki/Ordina...limit_ordinals):
If there is a point you would like to make, you make it. Again, your water, you carry it.

Quote:
Time after time the term "all" is used "out of the blue" without any reasoning behind it.
Out of the dictionary would be a more accurate statement.

Quote:
So, please demonstrate ZF reasoning, which actually consistently and rigorously establishes an infinite set that contains all of its members (for example: by ZF reasoning {0,1,2,3,4,5,...} is an infinite set that contains all of its members in spite of the fact that there is no largest natural number.
You keep bringing up that last bit as if it were somehow important. Mathematics is not bound to your lack of imagination about the characteristics of infinite sets.


By the way, all of this has been a nice dodge by you. You still have a task of defining your version of cardinality, a task that has so far proven impossible for you.
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Old 11th August 2020, 07:33 AM   #177
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Originally Posted by jsfisher View Post
Mathematics is not bound to your lack of imagination about the characteristics of infinite sets.
Then please demonstrate exactly how ZF consistently and rigorously establishes an infinite set that includes all of its members, in spite of my lack of imagination. (b.t.w since when imagination is a factor in your arguments about actually doing mathematics?)

Originally Posted by jsfisher View Post
Interesting bit of projection, there.
Well, unlike you, I do not claim that definitions can't be disproved in case that new notions about X are fundamentally changed.

Actually Cantor's approach about infinite sets was an attempt to fundamentally change our understanding about them.

ZF is an example of how to do math according to this attempt.

But when asked to consistently and rigorously use ZF in order to establish an infinite set that includes all of its members, you simply can't do that.

Instead, my lack of imagination is involved etc. ("all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied, exactly as the largest natural number does not exit, and no special kind of imagination is needed in order to understand it, just simple common sense (where being simple is far from being trivial)).

You claim that ZF consistently and rigorously establishes an infinite set that includes all of its members.

Your claim, your water; you carry it.
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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; 11th August 2020 at 07:50 AM.
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Old 11th August 2020, 08:22 AM   #178
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Originally Posted by doronshadmi View Post
....

All of this has been a nice dodge by you. You still have a task of defining your version of cardinality, a task that has so far proven impossible for you.
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Old 12th August 2020, 12:05 AM   #179
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D is a placeholder for any given set.

∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) )

Code:
V={
    
       ∅, 
       {∅}, 
       { ∅, {∅} }, 
       { ∅, {∅} , {∅, {∅}} }, 
       { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, 
...  

}

and it does not have the largest member.
Definition 1: Cardinality is the 'size' of D iff |D| is defined by V.

Definition 2: N is the set of finite cardinalities iff any given nN is defined by its corresponding V member.

For example:
Code:
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
...
Definition 3: D is called finite iff it is bijective with some particular V member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member since V does not have the largest member.


------------------------------------------------------------------------------


Originally Posted by jsfisher View Post
Which set would that be? Seems unlikely you would be referring to the ordinals recognized by von Neumann since there are considerably more of those than you'd like to admit.
Well, please show consistently and rigorously, how ω is established as a limit ordinal in spite of the equivalent fact that V and N do not have their largest members (it means that the term "all" in both sets is not satisfied, where this simple fact actually prevents the existence of ω as a size (called limit ordinal) that claimed to exist by traditional mathematics as follows:
Quote:
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
Such claim artificially establishes ω as a size larger than the sizes of "all" natural numbers, in spite of the fact that the term "all" is not satisfied, exactly because N does not have its largest member (ω as a size > "all" natural numbers, is not satisfied).

By using only imagination one can establish whatever he/she likes, by ignoring fundamental facts (as shown above) of the considered subject.

So the problem here, jsfisher, is not my "lack of imagination about the characteristics of infinite sets", but it is your inability to consistently and rigorously establish ω or |N|, exactly because your axiomatic framework (ZF in this case) artificially eliminates fundamental characteristics of infinite sets, where the lack of the term "all" is one of them.

"ω is called limit ordinal iff ω > all nN" is no more than a string of symbols based only on imagination, without any understanding of fundamental characteristics of N as a non-finite collection.
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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; 12th August 2020 at 12:57 AM.
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Old 12th August 2020, 02:35 AM   #180
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jsfisher, I wish you healthy and happy birthday.
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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 12th August 2020, 03:50 AM   #181
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Originally Posted by doronshadmi View Post
D is a placeholder for any given set.

∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) )
That is not a specific set. E.g., it is undetermined if {{∅}} is a member of V.

Quote:
Code:
V={
...  
}

and it does not have the largest member.
What does "largest member" mean?

Quote:
Definition 1: Cardinality is the 'size' of D iff |D| is defined by V.
(1) You still have that bogus "iff" in there. (2) How would |D| be defined by V? Isn't that what you are trying to define?
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Old 12th August 2020, 07:07 AM   #182
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Originally Posted by jsfisher View Post
That is not a specific set. E.g., it is undetermined if {{∅}} is a member of V.
This is a member of Zermelo ordinals.

Please demonstrate how this member (or any other member of Zermelo ordinals) is also one of the members of V, which are constructed in terms of von Neumann ordinals ( for any given x ∈ V ( ( x ∪ { x } ) ∈ V )).


Originally Posted by jsfisher View Post
What does "largest member" mean?
It means that V has the most complex member of von Neumann construction of sets.

But V does not have such member.


Originally Posted by jsfisher View Post
(1) You still have that bogus "iff" in there. (2) How would |D| be defined by V? Isn't that what you are trying to define?
V is defined by von Neumann construction of sets.

V has at least two basic properties:

(1) Any given member is finite.

(2) It does not have the most complex member of von Neumann construction of sets.

By (1) finite cardinality is defined in terms of natural numbers.

By (2) the search for some particular finite cardinality is (potentially) non-finite since the term "all" is not satisfied (our search continues endlessly among V members, and this is exactly the reason why ω as the limit ordinal of "all" smaller ordinals (which are V members) is not established.

Since ω is not established, the difference between cardinal numbers and ordinal numbers (as done by Cantor's Transfinite system, which forces sets to be defined in terms of actual infinity (for example: "The set of all natural numbers exists")) is not established.
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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; 12th August 2020 at 07:31 AM.
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Old 12th August 2020, 09:02 AM   #183
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Originally Posted by doronshadmi View Post
This is a member of Zermelo ordinals.

Please demonstrate how this member (or any other member of Zermelo ordinals) is also one of the members of V, which are constructed in terms of von Neumann ordinals ( for any given x ∈ V ( ( x ∪ { x } ) ∈ V )).
You presented two requirements for your set V. (1) It contains the empty set, and (2) for every member of V, V also contains that member's "successor".

Your requirements specify things the set V must have, but they do not identify anything as something the set must not have. They neither include nor exclude {{∅}} as a member of set V.

Quote:
It means that V has the most complex member of von Neumann construction of sets.
You are offering substitute words without conveying any meaning. What does "most complex member" mean?


For that matter, what bearing does having or not having a largest member have on this discussion? If there were a set which was both infinite and did have a "largest member" (by some appropriate definition of largest member), then what? What if there were set that had no largest member but was finite, then what?
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Old 12th August 2020, 12:44 PM   #184
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Posted only for doronshadmi's convenience (bolds and italics added):

I ( ∅ ∈ I ∧ ∀ xI ( ( x ∪ { x } ) ∈ I ) )
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Old 12th August 2020, 12:58 PM   #185
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Originally Posted by Little 10 Toes View Post
Posted only for doronshadmi's convenience (bolds and italics added):

I ( ∅ ∈ I ∧ ∀ xI ( ( x ∪ { x } ) ∈ I ) )
I always thought playing with the spacing a bit helped, too:
I (∅∈I ∧ ∀xI ((x∪{x})∈I))
[/pedant]
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Old 13th August 2020, 06:00 AM   #186
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Originally Posted by jsfisher View Post
You presented two requirements for your set V. (1) It contains the empty set, and (2) for every member of V, V also contains that member's "successor".

Your requirements specify things the set V must have, but they do not identify anything as something the set must not have. They neither include nor exclude {{∅}} as a member of set V.
For every member of V, V contains that member's "successor" that is defined by the union of x with {x} (written as x ∪ {x}).

Please define {{{}}} as the union of x with {x}.

Originally Posted by jsfisher View Post
You are offering substitute words without conveying any meaning. What does "most complex member" mean?
As seen in case of V members' construction, they become more and more complex (where the least complex member is {}), yet V does not have the most complex member, by this members' construction.

Originally Posted by jsfisher View Post
For that matter, what bearing does having or not having a largest member have on this discussion? If there were a set which was both infinite and did have a "largest member" (by some appropriate definition of largest member), then what?
Please define such set, but please remember that its members are pure sets (constructed only by "{"..."}" pairs, at the base level of sets' representations).


Originally Posted by jsfisher View Post
What if there were set that had no largest member but was finite, then what?
Again, please think about non-empty sets that its distinct members are pure sets (constructed only by "{"..."}" pairs, at the base level of sets' representations).

Originally Posted by jsfisher View Post
The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function. It merely states two properties the set has (i.e. that it contains the empty set and that every member of the set also has its successor as a member).

The Axiom is silent on whether, for example, {{{{ }}}} is in the set.

The Axiom must to be coupled with other axioms to conclude von Neumann's ordinal is a set in ZF.
By ∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) ) we get V members.

By ∃ X ( ∅ ∈ X ∧ for any given x ∈ X ( { x } ∈ X ) ) we don't get V members (except ∅ that is not a successor in both constructions, and it is specifically included in both sets).

You see jsfisher, you take V as a placeholder for any set, but V has a unique construction, which is different than X unique construction.

Please look at http://www.internationalskeptics.com...&postcount=179 .

As you can see there, D is a placeholder for any set, but V is a unique set that is used to define cardinality, whether it is finite or (potentially) non-finite (since its largest member does not exist).
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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; 13th August 2020 at 07:06 AM.
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Old 13th August 2020, 08:44 AM   #187
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Originally Posted by doronshadmi View Post
For every member of V, V contains that member's "successor" that is defined by the union of x with {x} (written as x ∪ {x}).

Please define {{{}}} as the union of x with {x}.
The requirement you have cited does not rule out {{∅}} from being in V. It does guarantee that if {{∅}} be a member, then so must { {{∅}}, {{{∅}}} }, but it is silent as to whether {{∅}} is a member in the first place.

Quote:
As seen in case of V members' construction....
You use the term, construction, as if the set comes into existence by a sequence of steps. It does not.
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Old 13th August 2020, 05:17 PM   #188
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Originally Posted by jsfisher View Post
I always thought playing with the spacing a bit helped, too:
I (∅∈I ∧ ∀xI ((x∪{x})∈I))
[/pedant]
I prefer the other one. It's harder to distinguish symbols like } and ) when they are right next to each other than when they are spaced a bit apart. I also find it easier to tell the structure of a proposition when it's spaced out a bit.
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Old 13th August 2020, 06:57 PM   #189
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Originally Posted by caveman1917 View Post
I prefer the other one. It's harder to distinguish symbols like } and ) when they are right next to each other than when they are spaced a bit apart. I also find it easier to tell the structure of a proposition when it's spaced out a bit.
Yeah, that is all true. The default font size this forum uses doesn't help much either.

Here it is at SIZE = 5: I (∅∈I ∧ ∀xI ((x∪{x})∈I))

I also considered substituting brackets for parenthesis at one place to alternate the grouping symbols.

Something like this: I (∅∈I ∧ ∀xI [(x∪{x})∈I])

The bold isn't all that helpful at SIZE = 5, but braces are more obvious.

Switching to Times New Roman: I (∅∈I ∧ ∀xI [(x∪{x})∈I])


YMMV
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Last edited by jsfisher; 13th August 2020 at 07:02 PM.
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Old 13th August 2020, 08:13 PM   #190
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Originally Posted by caveman1917 View Post
I prefer the other one. It's harder to distinguish symbols like } and ) when they are right next to each other than when they are spaced a bit apart. I also find it easier to tell the structure of a proposition when it's spaced out a bit.
Never thought I would ever say this about me....






The Nominate button is right down there...
vvvvvvvv
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Old 13th August 2020, 08:16 PM   #191
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I notice doronshadmi dodged at least two direct questions:
Originally Posted by jsfisher View Post
You are offering substitute words without conveying any meaning. What does "most complex member" mean?
and
Quote:
For that matter, what bearing does having or not having a largest member have on this discussion? If there were a set which was both infinite and did have a "largest member" (by some appropriate definition of largest member), then what? What if there were set that had no largest member but was finite, then what?
(original quote edited)

Is there a reason you can't/won't answer these questions doronshadmi?
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Old 14th August 2020, 04:02 AM   #192
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Originally Posted by jsfisher View Post
The requirement you have cited does not rule out {{∅}} from being in V. It does guarantee that if {{∅}} be a member, then so must { {{∅}}, {{{∅}}} }, but it is silent as to whether {{∅}} is a member in the first place.
∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) ) has the following members according to its given construction:
Code:
V={
    
       ∅, 
       {∅}, 
       { ∅, {∅} }, 
       { ∅, {∅} , {∅, {∅}} }, 
       { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, 
... 

}
∃ X ( ∅ ∈ X ∧ for any given x ∈ X ( { x } ∈ X ) ) has the following members according to its given construction:
Code:
X={
    
       ∅, 
       {∅}, 
       {{∅}}, 
       {{{∅}}}, 
       {{{{∅}}}}, 
... 

}

Originally Posted by jsfisher View Post
You use the term, construction, as if the set comes into existence by a sequence of steps. It does not.
No steps (from the Platonist point of view) the unique construction of V or X sets is taken at once (or in parallel, if you wish).

From the, so called, constructionist point of view, it may be taken serially.

Some common properties of both V and X sets are:

1) No successor of x is a member of x (in order to be considered as a successor, in the first place)

2) Any given member of X or Y is a finite set.
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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; 14th August 2020 at 06:00 AM.
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Old 14th August 2020, 06:06 AM   #193
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Some corrections of my previous post:

2) Any given member of V or X is a finite set.

3) No successor is the largest member of V or X.
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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 14th August 2020, 06:12 AM   #194
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Originally Posted by jsfisher View Post
Yeah, that is all true. The default font size this forum uses doesn't help much either.

Here it is at SIZE = 5: I (∅∈I ∧ ∀xI ((x∪{x})∈I))

I also considered substituting brackets for parenthesis at one place to alternate the grouping symbols.

Something like this: I (∅∈I ∧ ∀xI [(x∪{x})∈I])

The bold isn't all that helpful at SIZE = 5, but braces are more obvious.

Switching to Times New Roman: I (∅∈I ∧ ∀xI [(x∪{x})∈I])


YMMV
Yes if you increase the font size your version is better as it "groups" terms together. But with the default font size putting symbols right next to each other makes it less readable.
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Old 14th August 2020, 06:13 AM   #195
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Originally Posted by Little 10 Toes View Post
Never thought I would ever say this about me....






The Nominate button is right down there...
vvvvvvvv
Do mathematical languages count for the language awards? Like nominating your post for the most readable rendition of the existence of N without needing to increase font size?
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Old 14th August 2020, 09:10 AM   #196
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Originally Posted by doronshadmi View Post
∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) )
That would be the Axiom on Infinity. It stipulates the existence of a set (that's the ∃V part) with two properties. It does not, however, tell us specifically what that set is.

Quote:
...has the following members according to its given construction
No, there no construction, neither step-by-step nor all at once.


Doronshadmi, I have a bag of marbles. All of my blue marbles are in the bag. Is there a red marble in the bag?

Answer: From the information I've given you, you cannot tell.

Nor can you tell if {{∅}} be a member of the set stipulated by the Axiom of Infinity.
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Old 15th August 2020, 09:05 PM   #197
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I notice doronshadmi dodged at least two direct questions:
Originally Posted by jsfisher View Post
You are offering substitute words without conveying any meaning. What does "most complex member" mean?
and
Quote:
For that matter, what bearing does having or not having a largest member have on this discussion? If there were a set which was both infinite and did have a "largest member" (by some appropriate definition of largest member), then what? What if there were set that had no largest member but was finite, then what?
(original quote edited)

Is there a reason you can't/won't answer these questions doronshadmi, yet again?
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Old 16th August 2020, 06:22 AM   #198
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Originally Posted by jsfisher View Post
I apologize. It seems I have introduced some confusion I did not intend by using the term "von Neumann ordinal" in a non-standard way.

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.
∃ N ( ∅ ∈ N ∧ for any given x ∈ N ( ( x ∪ { x } ) ∈ N ) )

Since "base elements [predecessors], and their successors" are defined by the axiom of infinity, N members are ordered, otherwise the term successor is not used, in the first place.

Moreover, x is the immediate predecessor of ( x ∪ { x } ) and ( x ∪ { x } ) is the immediate successor of x, so the way of how any given non-empty x member is defined, depends on how the successor of any given non-empty x is defined (they are its distinct foot prints that it leaves 'behind').

Definition 1: Set Y is called the immediate successor of set X iff X is the set of all the predecessors of Y AND Y is not a member of X (where the term all is valid only if X is a finite set, as will be shown in the case of set N).

So set N has ∅ as its base member, which is the predecessor of any given non-empty set that is a member of set N.

Since order is important (otherwise the terms predecessor or successor are not used, in the first place) N members (by the axiom of infinity) are as follows:

Code:
N={
    
       ∅ (the base member), 
       ∅∪{∅}={∅} , 
       {∅}∪{{∅}}={ ∅, {∅} }, 
       {∅,{∅}}∪{{∅,{∅}}}={ ∅, {∅} , {∅, {∅}} }, 
       {∅,{∅},{∅,{∅}}}∪{{∅,{∅},{∅,{∅}}}}={ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, 
... 

}
So as can clearly be seen, Zermelo sets (except the first two members) are not defined by ZF axiom of infinity.

-------------

(B.T.W the largest member of a given set, is the member of that set, which has no 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.

Last edited by doronshadmi; 16th August 2020 at 07:10 AM.
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Old 16th August 2020, 06:44 AM   #199
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doronshadmi, if you are going to use the Axiom of Infinity, quote it correctly. This shows your level of credibility. It has been provided to you several times on this page alone.

∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I])
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Old 16th August 2020, 10:09 AM   #200
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Originally Posted by doronshadmi View Post
Since "base elements [predecessors]...are defined by the axiom of infinity....
No, they are not. The Axiom of Infinity requires the null set be in the set. It says nothing about other possibilities. The Axiom does not identify a specific set.
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