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Old 29th August 2020, 04:09 AM   #281
doronshadmi
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Also please be aware that if N is one of its base members, then N is of the form (x∪{x}) (a successor) that does not have its largest successor, since ∀ is not a property of N.

More details are seen in http://www.internationalskeptics.com...&postcount=280 .
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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 August 2020 at 04:14 AM.
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Old 29th August 2020, 06:45 AM   #282
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Originally Posted by Little 10 Toes View Post
Let's make it easy since doronshadmi is not understanding the simple items.

Which, if any, of the following items are in N?
1) The War of 1812
2) 10% GDP
3) Oranges
4) The ratio between the circumference of a circle and it's diameter
5) Beethoven's Fifth
6) {... -2, -1, 0, 1, 2, ...}
7) { {∅} ,{ { { {∅} } } } }
8) Double shot of espresso, 3 soy creams, 1 brown sugar, 97 degree, no whip

Please note, since doronshadmi has used N before, I am confirming that I am not talking about the set of natural numbers (also known/written as N or N).
Doronshadmi, do you understand that all of these items can be elements of set N?
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Old 29th August 2020, 08:37 AM   #283
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Originally Posted by doronshadmi View Post
Originally Posted by jsfisher View Post
You have been outspoken that there is no largest successor. That means that any set I choose is going to be "~largest successor". That means that any set I choose (including N) is going to be a member of N.
N does not have even one largest successor, this is exactly why N can't be but an infinite set that includes base sets OR successor sets that are not largest successors sets.
You didn't understand my post at all, did you? Your response is just trying to cycle back to your hatred of the universal quantifier without regard to the topic at hand.

You so-called membership function is bogus. About your so-called membership function you wrote:
Originally Posted by doronshadmi View Post
x is largest successor is the same as x is not a member.

x is ~largest successor is the same as x is a member.
Well, guess what? Your set N is a set, and N is not the largest successor of N. Therefore, by your so-called membership function, N must be a member of N.
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Old 29th August 2020, 08:42 AM   #284
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Originally Posted by doronshadmi View Post
Also please be aware that if N is one of its base members, then N is of the form (x∪{x}) (a successor) that does not have its largest successor, since ∀ is not a property of N.

That's nice, but it has nothing to do with you so-called membership function, M.

Originally Posted by doronshadmi View Post
If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.
Clearly, M(N) must be true, so N is a member of N.
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Old 29th August 2020, 09:22 AM   #285
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Originally Posted by jsfisher View Post
Clearly, M(N) must be true, so N is a member of N.
N as a member of N is of the form (x∪{x}) (a successor) that does not have its largest successor, since ∀ is not a property of N.

If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.

Since N is a successor that is not largest successor, then M(N) is indeed true, since ∀ is not a property of N.
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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 August 2020, 10:12 AM   #286
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Originally Posted by doronshadmi View Post
N as a member of N is of the form (x∪{x})

If that's what you want, then you need to include that in your membership function directly. As written every set is a member of N. That's not a very good membership function.
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Old 29th August 2020, 11:02 AM   #287
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Originally Posted by jsfisher View Post
As written every set is a member of N. That's not a very good membership function.
Not every set, a successor set that is also largest set is not a member of set N.

So terms like "every set", "all sets", ∀ etc. are not properties of set N.
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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 August 2020, 11:11 AM   #288
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Originally Posted by doronshadmi View Post
Not every set, a successor set that is also largest set is not a member of set N.
Well, since there is no "largest successor set", it is every set. Your N is the set of all sets, or at least that is what your membership function would have us believe.

Your membership function is nonsense. You haven't gotten any closer to defining your set N. Defining N is key to what you think you need to define cardinality, so that sits dead on the vine as well.

You have made no progress on the topic at hand.
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Old 29th August 2020, 11:15 AM   #289
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Originally Posted by jsfisher View Post
If that's what you want, then you need to include that in your membership function directly.
It is included directly:

If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.

Not being largest successor set is:

(successor set AND not largest successor set) OR (base set).

In case of N as a member of N, N is a (successor set AND not largest successor set), which means that "every set", "all sets", ∀ etc. are not properties of set N.
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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 August 2020, 11:19 AM   #290
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Originally Posted by jsfisher View Post
Well, since there is no "largest successor set", it is every set. Your N is the set of all sets,
EDIT:
Wrong again, jsfisher, terms like "every set", "all sets", ∀ etc. are not properties of set N, exactly because any one of its successor sets is (successor set AND not largest successor 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; 29th August 2020 at 11:23 AM.
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Old 29th August 2020, 11:30 AM   #291
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Originally Posted by doronshadmi View Post
Not being largest successor set is:

(successor set AND not largest successor set) OR (base set)
...deleted...

I want to rewrite this differently.
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Last edited by jsfisher; 29th August 2020 at 11:40 AM.
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Old 29th August 2020, 11:34 AM   #292
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Originally Posted by doronshadmi View Post
EDIT:
Wrong again, jsfisher, terms like "every set", "all sets", ∀ etc. are not properties of set N, exactly because any one of its successor sets is (successor set AND not largest successor set).

"∀" is not now nor has it ever been a property. You've proven again you don't understand the meaning of basic mathematical concepts. You are in no position to lash out against them. Please stop pretending.
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Old 29th August 2020, 11:44 AM   #293
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EDIT:
Originally Posted by jsfisher View Post
"(successor set AND not largest successor set) OR (base set)" is universally true.
No, take for example a finite set of the form (x∪{x}). It has largest successor set, something that N does not have.

Originally Posted by jsfisher View Post
"∀" is not now nor has it ever been a property. You've proven again you don't understand the meaning of basic mathematical concepts. You are in no position to lash out against them. Please stop pretending.
Therefore "every set", "all sets", ∀ etc. can't be related to set N, exactly because it is an infinite 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; 29th August 2020 at 11:50 AM.
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Old 29th August 2020, 12:26 PM   #294
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Originally Posted by doronshadmi View Post
It is included directly:

If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.

Not being largest successor set is:

(successor set AND not largest successor set) OR (base set).

Your most recent set of supporting definitions are these:

Originally Posted by doronshadmi View Post
Let's put what is needed in one post, in order to define N as a set.

Definition 1: Any given set that is not of the form (x∪{x}), is called a base set.

Definition 2: Any given set that is of the form (x∪{x}), is called a successor set.

Definition 3: Given a successor set in some set, it is called largest successor set iff given a base set in that set, it has at least one successor set that does not have its successor set, in that set.
You never qualified "any given set" to be restricted to being in N. Did you mean to? You never stated how one would determine if this "any given set" is of the form x ∪ {x}. How would you?

As for definition 3, well, it is badly worded and doesn't state what you think it does. What are you trying to say? Whatever you meant by it, it certainly was not this:
Not being largest successor set is:
(successor set AND not largest successor set) OR (base set).
If nothing else, the thing you are trying to define (largest successor set) appears in the definition.

Remember, too, the goal you are trying to attain for the present is a definition for your set N. Something like:
N = { y : M(y) }
with a formulation for M acceptable to the set theory axioms. Your so-called definitions are little more than a distraction from the goal.
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Old 29th August 2020, 12:29 PM   #295
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Originally Posted by doronshadmi View Post
Therefore "every set", "all sets", ∀ etc. can't be related to set N, exactly because it is an infinite set.
Not a relation either. Please stop pretending.
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Old 29th August 2020, 11:11 PM   #296
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Originally Posted by jsfisher View Post

As for definition 3, well, it is badly worded and doesn't state what you think it does. What are you trying to say? Whatever you meant by it, it certainly was not this:
Not being largest successor set is:
(successor set AND not largest successor set) OR (base set).
If nothing else, the thing you are trying to define (largest successor set) appears in the definition.
"Not being largest successor" set is not the definition of largest successor set, but it is the use of the term largest set as part of the expression (successor set AND not largest successor set) "after" it is defined.

You are mixing between the use of the term "largest successor set" in some expression ("after" it is defined) and the definition of the term "largest successor set" which is:

Definition 3: Given a successor set in some set, it is called largest successor set iff given a base set in that set, it has at least one successor set that does not have its successor set, in that set.

What is written after the iff does not use the term "largest successor set" in order to define "largest successor set".

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

I agree with you that the expression (successor set AND not largest successor set) is misleading, so let's correct it as follows:


If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.

Not being largest successor set in N is:

(successor set that has its successor set) OR (base set).

In case of N as a member of N, N is a (successor set that has its successor set), which means that "every set", "all sets", ∀ etc. are not related to set N.
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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 August 2020 at 11:45 PM.
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Old 30th August 2020, 12:01 AM   #297
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Originally Posted by jsfisher
You never stated how one would determine if this "any given set" is of the form x ∪ {x}. How would you?
Please demonstrate exactly how one can't determine if a given set is of the form x ∪ {x} or not of the form x ∪ {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 30th August 2020, 12:09 AM   #298
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Originally Posted by jsfisher View Post
Not a relation either. Please stop pretending.
Relation is a noun, related is a verb.

I used "related".

If you have a better word of the connection between ∀ and a set, then please say 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; 30th August 2020 at 12:20 AM.
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Old 30th August 2020, 07:43 AM   #299
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Originally Posted by jsfisher View Post
You never stated how one would determine if this "any given set" is of the form x ∪ {x}. How would you?
Originally Posted by doronshadmi View Post
Please demonstrate exactly how one can't determine if a given set is of the form x ∪ {x} or not of the form x ∪ {x}.
  1. Have the common decency to use the quote function properly to refer to the message that you are quoting from. I have done that for you with jsfisher's original message. You're welcome.
  2. How about you stop shifting the burden. Your claim, your evidence. You show how you would determine "any given set" is of the form x ∪ {x}.

Also:
1) The War of 1812
2) 10% GDP
3) Oranges
4) The ratio between the circumference of a circle and it's diameter
5) Beethoven's Fifth
6) {... -2, -1, 0, 1, 2, ...}
7) { {∅} ,{ { { {∅} } } } }
8) Double shot of espresso, 3 soy creams, 1 brown sugar, 97 degree, no whip

Do you understand that all of these items can be elements of set N?

Since this is the third time that I am asking, I am going to assume that you can't.
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Old 30th August 2020, 08:39 AM   #300
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Originally Posted by doronshadmi View Post
"Not being largest successor" set is not the definition of largest successor set, but it is the use of the term largest set as part of the expression (successor set AND not largest successor set) "after" it is defined.
You said: "Not being largest successor set is...". So, yeah, it is a definition (albeit in the negative).
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Old 30th August 2020, 08:45 AM   #301
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Originally Posted by doronshadmi View Post
Please demonstrate exactly how one can't determine if a given set is of the form x ∪ {x} or not of the form x ∪ {x}.
I didn't say one couldn't determine if a set has a certain structure; I am asking you to clearly state how can be determined.
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Old 30th August 2020, 08:50 AM   #302
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Originally Posted by doronshadmi View Post
Relation is a noun, related is a verb.

I used "related".

If you have a better word of the connection between ∀ and a set, then please say it.
Not my responsibility.

Be that as it may, how about getting back to the task at hand: Clean up your definitions and define this set N of yours.
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Old 31st August 2020, 04:46 AM   #303
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Originally Posted by jsfisher View Post
You said: "Not being largest successor set is...". So, yeah, it is a definition (albeit in the negative).
Please read all of http://www.internationalskeptics.com...&postcount=296 about this case, it was corrected according to your criticism, but somehow you skipped on 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.
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Old 31st August 2020, 05:01 AM   #304
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Originally Posted by jsfisher View Post
I didn't say one couldn't determine if a set has a certain structure; I am asking you to clearly state how can be determined.
Please look at ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]).

Exactly as given in this axiom, any given set that is of the form (x∪{x}), can't be but of the form (x∪{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.

Last edited by doronshadmi; 31st August 2020 at 05:06 AM.
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Old 31st August 2020, 06:18 AM   #305
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Originally Posted by doronshadmi View Post
Please look at ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]).

Exactly as given in this axiom, any given set that is of the form (x∪{x}), can't be but of the form (x∪{x}).
That is of no use to you. You need a function, let's call it IsSucc(y), that is true if and only if y is of the required form. Then and only then can you clean up your definitions to be something like this:
Definition: A set X is a successor set iff IsSucc(X).
Definition: A set X is a base set iff it is not a successor set.
Neither definition is all that important, by the way, since you can just use the IsSucc() function directly as you formulate your membership function for N.
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Old 31st August 2020, 06:59 AM   #306
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and remember, these are not computer science functions, but math functions.
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Old 31st August 2020, 07:34 AM   #307
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OK, I see what you mean, for example:

{} ∪ any non-empty set is that non-empty set, so the form (x∪{x}) can't be used in order to distinguish between what I call base set and successor set.

In that case I have to define what is the distinction between base set and a successor set, in the first place.

In other words, I have no framework until I define successor.

Thank you jsfisher.

So, back to work.
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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 31st August 2020, 07:52 AM   #308
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Originally Posted by Little 10 Toes View Post
  1. Have the common decency to use the quote function properly to refer to the message that you are quoting from. I have done that for you with jsfisher's original message. You're welcome.
  2. How about you stop shifting the burden. Your claim, your evidence. You show how you would determine "any given set" is of the form x ∪ {x}.

Also:
1) The War of 1812
2) 10% GDP
3) Oranges
4) The ratio between the circumference of a circle and it's diameter
5) Beethoven's Fifth
6) {... -2, -1, 0, 1, 2, ...}
7) { {∅} ,{ { { {∅} } } } }
8) Double shot of espresso, 3 soy creams, 1 brown sugar, 97 degree, no whip

Do you understand that all of these items can be elements of set N?

Since this is the third time that I am asking, I am going to assume that you can't.
Little 10 Toes, the current discussion is only about sets.

For example 1 or 8 can be members of sets, but they are not themselves sets (they are called urelements ( see, for example, in https://en.wikipedia.org/wiki/Urelement )).
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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 31st August 2020, 09:15 AM   #309
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Originally Posted by doronshadmi View Post
Little 10 Toes, the current discussion is only about sets.

For example 1 or 8 can be members of sets, but they are not themselves sets (they are called urelements ( see, for example, in https://en.wikipedia.org/wiki/Urelement )).
No, the current discussion is that you currently can't define sets.

Here is a list of elements:
1) The War of 1812
2) 10% GDP
3) Oranges
4) The ratio between the circumference of a circle and it's diameter
5) Beethoven's Fifth
6) {... -2, -1, 0, 1, 2, ...}
7) { {∅} ,{ { { {∅} } } } }
8) Double shot of espresso, 3 soy creams, 1 brown sugar, 97 degree, no whip

Do you understand that all of these items can be elements of a set?

Since this is the fourth that I am asking, I am going to assume that you can't.
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Old 1st September 2020, 08:27 AM   #310
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Originally Posted by doronshadmi View Post
{} ∪ any non-empty set is that non-empty set, so the form (x∪{x}) can't be used in order to distinguish between what I call base set and successor set.
Ok, no more jokes, it is easy to understand that what is written above is nonsense, since in case of x={} the form (x∪{x}) is the particular case {}∪{{}} = {{}} that is the successor of {}.

In other words, any given set of the form (x∪{x}) can't be but the successor set of any given set x.

This exactly the reason why in ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) the form (x∪{x}) appears whiteout the need to first define it as a successor.

So the following quote
Originally Posted by jsfisher
Definition: A set X is a successor set iff IsSucc(X).
Definition: A set X is a base set iff it is not a successor set.
Is actually
Definition 1: A set X is a successor set iff (x∪{x}).
Definition 2: A set X is a base set iff ~(x∪{x}).
Definition 3: Given a successor set in some set, it is called largest successor set iff given a base set in that set, it has at least one successor set that does not have its successor set, in that set.
Originally Posted by jsfisher
Neither definition is all that important, by the way, since you can just use the IsSucc() function directly as you formulate your membership function for N.
Set N members are (x∪{x}) or ~(x∪{x}), such that ~(x∪{x}) is a base set and (x∪{x}) is a successor set, where given any successor set, it is not largest successor set, since unlike definition 3, its successor set is a member of N.

Moreover, if x=N, then N is not the same as (N∪{N}), and we can't claim that N is a member of N, exactly as we can't claim that, for example, {} is a member of itself, since {}∪{{}}={{}} not={}.

In other words, ∀ and N have nothing to do with each other, which enables to define N as a set of infinitely many members that its cardinality is not any particular size.
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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 1st September 2020, 09:02 AM   #311
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Originally Posted by doronshadmi View Post
Definition 1: A set X is a successor set iff (x∪{x}).
(x ∪ {x}) is not a boolean expression; it is a set. Did you mean X = (x ∪ {x})? If so, then what is this lower-case x? How is this lower-case x identified?
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Old 1st September 2020, 09:16 AM   #312
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Originally Posted by doronshadmi View Post
This exactly the reason why in ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) the form (x∪{x}) appears whiteout the need to first define it as a successor.
No, the formula (x ∪ {x}) appears without "the need to first define it as a successor" because it is unnecessary to give it a name. That is what your first two still-in-need-of-work definitions attempt do: name things unnecessarily.
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Old 2nd September 2020, 01:18 AM   #313
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Originally Posted by jsfisher View Post
(x ∪ {x}) is not a boolean expression; it is a set. Did you mean X = (x ∪ {x})? If so, then what is this lower-case x? How is this lower-case x identified?
Definition 1: A set X is a successor set iff (x∪{x}).

Definition 2: A set X is a base set iff ~(x∪{x}).

x appears in two forms (~(x ∪ {x}) or (x ∪ {x})) under unification that provides the members of set X, if X is defined by definition 1.

For example:


x={} (it is ~(x ∪ {x}) and therefore it is a base set).

In that case X is the unification of the "members" of {} and the members of {{}}, so X={{}}, which is the successor set of set x.


x={{}} (it is (x ∪ {x}) and therefore it is a successor set).

In that case X is the unification of the members of {{}} and the members of {{{}}}, so X={{},{{}}}, which is the successor set of set x.


x={{},{{}}} (it is (x ∪ {x}) and therefore it is a successor set).

In that case X is the unification of the members of {{},{{}}} and the members of {{{},{{}}}}, so X={{},{{}},{{},{{}}}}, which is the successor set of set x.


x={{},{{}},{{},{{}}}}

etc. ad infinitum ...

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

x={{{}}}

and we have a base set and successor sets that are distinct from the sets in the example above, ad infinitum ...

etc. ad infinitum ...

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


Definition 2: A set X is a base set iff ~(x∪{x}).

In that case X is, for example, {}, {{{}}}, etc. that are not of the form (x∪{x}).
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That is also over the matrix, is aware of the matrix.

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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.

Last edited by doronshadmi; 2nd September 2020 at 03:00 AM.
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Old 2nd September 2020, 01:40 AM   #314
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Originally Posted by jsfisher View Post
No, the formula (x ∪ {x}) appears without "the need to first define it as a successor" because it is unnecessary to give it a name.
This is the reason why you do not distinguish between base sets ( ~(x ∪ {x}) ) and successor sets ( (x ∪ {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.

Last edited by doronshadmi; 2nd September 2020 at 01:52 AM.
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Old 2nd September 2020, 02:20 AM   #315
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X is ~(x∪{x}) OR (x∪{x}) and so is 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 2nd September 2020, 03:50 AM   #316
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Originally Posted by doronshadmi View Post
Definition 1: A set X is a successor set iff (x∪{x}).
Repeating this doesn't make them into a well-formed formula. The expression, x ∪ {x}, continues to be a set and not logical-valued.

You need something that is true or false on the right-hand side of if-and-only-if.
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Old 2nd September 2020, 04:13 AM   #317
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The definitions without iff.

Let's research ∃N (∅∈N ∧ ∀x∈N [(x∪{x})∈N]).

Definition 1: Any given set that is not of the form (x∪{x}), is called a base set.

Definition 2: Any given set that is of the form (x∪{x}), is called a successor set.

Definition 3: Given a successor set in some set, it is called largest successor set iff given a base set in that set, it has at least one successor set that does not have its successor set, in that set.

Let M(x) be the membership function of set N.

If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.

Not being largest successor set in N is:

(successor set that has its successor set) OR (base set)

An example:

Code:
N = {
∅,                    ---> {{∅}}, ---> ...   <-- base sets
{∅},                  ---> {{∅}, {{∅}} }, ---> ...  <-- successor sets
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...  <-- successor sets
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...  <-- successor sets
...                         ...
}
As can be seen, set N members are of the form ~(x∪{x}) (base sets) OR (x∪{x}) (successor sets that have their successor sets) (M(x) is true), but no largest successor set is a member of set N (M(x) is false).

Set N members are (x∪{x}) or ~(x∪{x}), such that ~(x∪{x}) is a base set and (x∪{x}) is a successor set, where given any successor set, it is not largest successor set, since unlike definition 3, its successor set is a member of N.

Moreover, if x=N, then N is not the same as (N∪{N}), and we can't define N as its own member, exactly as we can't define, for example, {} as its own member, since {}∪{{}}={{}}≠{}.

In other words, ∀ and N have nothing to do with each other, which enables to define N as a set of infinitely many members that its cardinality is not any particular size.

Originally Posted by jsfisher View Post
No, the formula (x ∪ {x}) appears without "the need to first define it as a successor" because it is unnecessary to give it a name. That is what your first two still-in-need-of-work definitions attempt do: name things unnecessarily.
Without define the formula (x∪{x}) as a successor set, the following is actually unknown:

Any set that is the member of itself is actually a successor set of that set, for example:

x={a,b,c,...}

If {a,b,c,...} is a member of itself, then we get the set {a,b,c,...{a,b,c,...}} which is actually ({a,b,c,...}∪{{a,b,c,...}}) = (x∪{x}) ≠ x = {a,b,c,...}

More general, no set is its successor set.


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

jsfisher, another unknown thing by mathematicians that do not define the formula (x∪{x}) as a successor set, is ∀ as the cause of Russell's Paradox, and this time please do not skip on it.


∀ is the cause of Russell's Paradox, whether a given collection of distinct objects is finite, or not.

For example:

U is a set of two distinct members, such that one of the members, called u, shaves ∀ the members of set U that do not shave themselves and only these members of set U (this is supposed to be his property in order to be a member of set U).

Who shaves u?

If u shaves himself, then he must not shave himself (shaves AND ~shaves himself, which is a contradiction) exactly because of the term ∀.

If u does not shave himself, then he must shave himself (~shaves AND shaves himself, which is a contradiction) exactly because of the term ∀.

So, because the term ∀ is used as a part of the terms that define u as a member U, u must be referred to himself, and we get the contradictions that actually prevents to well-define ∀ the members of set U (the term ∀ itself is actually not well-defined in case of U).

The same problem holds also among infinite sets that the term ∀ is one of their properties, therefore the Axiom of Restricted Comprehension was add to ZF in order to avoid Russell's Paradox, but it is done without being aware of the fact that the term ∀ is the cause of any given contradictory self reference, whether it is used among finite or infinite sets.
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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; 2nd September 2020 at 06:00 AM.
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Old 2nd September 2020, 11:13 AM   #318
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Originally Posted by doronshadmi View Post
Let's research ∃N (∅∈N ∧ ∀x∈N [(x∪{x})∈N]).
That is the Axiom of Infinity. It does not define any set; it simply declares the existance of some set that satisfies two properties. In particular, it does not define your set, N, so let's go with this version of the Axiom
∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])
to avoid any confusion.

Quote:
Definition 1: Any given set that is not of the form (x∪{x}), is called a base set.
For any given set, let's call it K, how do you determine if K is of the form x ∪ {x}?
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Old 2nd September 2020, 11:02 PM   #319
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Originally Posted by jsfisher View Post
That is the Axiom of Infinity. It does not define any set; it simply declares the existance of some set that satisfies two properties. In particular, it does not define your set, N, so let's go with this version of the Axiom
∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])
to avoid any confusion.



For any given set, let's call it K, how do you determine if K is of the form x ∪ {x}?
I take the property (x ∪ {x}) right from the axiom above, as follows:

Definition 1: A set K is a base set iff K property is ~(x∪{x}).

Definition 2: A set K is a successor set iff K property is (x∪{x}).

Each one of the given definitions simply declares the existence of some set that satisfies a given property.

Definition 3: A set K is a largest successor set in X iff K∈X AND K successor set ∉ X.

Let M(x) be the membership function of set N.

If x is largest successor, then M(x) : false.

If x is not largest successor, then M(x) : true.

Not being largest successor set in N is:

(successor set that has its successor set) OR (base set)

An example:

Code:
N = {
∅,                    ---> {{∅}}, ---> ...   <-- base sets
{∅},                  ---> {{∅}, {{∅}} }, ---> ...  <-- successor sets
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...  <-- successor sets
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...  <-- successor sets
...                         ...
}
As can be seen, set N members are of the form ~(x∪{x}) (base sets) OR (x∪{x}) (successor sets that have their successor sets) (M(x) is true), but no largest successor set is a member of set N (M(x) is false).

Set N members are (x∪{x}) or ~(x∪{x}), such that ~(x∪{x}) is a base set and (x∪{x}) is a successor set, where given any successor set, it is not largest successor set, since unlike definition 3, its successor set is a member of N.

Moreover, if x=N, then N is not the same as (N∪{N}), and we can't define N as its own member, exactly as we can't define, for example, {} as its own member, since {}∪{{}}={{}}≠{}.

In other words, ∀ and N have nothing to do with each other, which enables to define N as a set of infinitely many members that its cardinality is not any particular size.



Without definition 2, the following is actually unknown:

Any set that is the member of itself is actually a successor set of that set, for example:

x={a,b,c,...}

If {a,b,c,...} is a member of itself, then we get the set {a,b,c,...{a,b,c,...}} which is actually ({a,b,c,...}∪{{a,b,c,...}}) = (x∪{x}) ≠ x = {a,b,c,...}

More general, no set is its successor set.


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

jsfisher, another unknown thing by mathematicians that do not define successor set, is ∀ as the cause of Russell's Paradox, and this time please do not skip on it.


∀ is the cause of Russell's Paradox, whether a given collection of distinct objects is finite, or not.

For example:

U is a set of two distinct members, such that one of the members, called u, shaves ∀ the members of set U that do not shave themselves and only these members of set U (this is supposed to be his property in order to be a member of set U).

Who shaves u?

If u shaves himself, then he must not shave himself (shaves AND ~shaves himself, which is a contradiction) exactly because of the term ∀.

If u does not shave himself, then he must shave himself (~shaves AND shaves himself, which is a contradiction) exactly because of the term ∀.

So, because the term ∀ is used as a part of the terms that define u as a member U, u must be referred to himself, and we get the contradictions that actually prevents to well-define ∀ the members of set U (the term ∀ itself is actually not well-defined in case of U).

The same problem holds also among infinite sets that the term ∀ is one of their properties, therefore the Axiom of Restricted Comprehension was add to ZF in order to avoid Russell's Paradox, but it is done without being aware of the fact that the term ∀ is the cause of any given contradictory self reference, whether it is used among finite or infinite sets.
__________________
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; 3rd September 2020 at 12:16 AM.
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Old 3rd September 2020, 02:39 AM   #320
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Some improvement of definition 3:

Definition 3: A successor set K is a largest successor set in X iff K∈X AND K successor set ∉ 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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