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29th August 2020, 04:09 AM  #281 
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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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29th August 2020, 06:45 AM  #282 
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29th August 2020, 08:37 AM  #283 
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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 socalled membership function is bogus. About your socalled membership function you wrote: Well, guess what? Your set N is a set, and N is not the largest successor of N. Therefore, by your socalled membership function, N must be a member of N. 
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29th August 2020, 08:42 AM  #284 
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29th August 2020, 09:22 AM  #285 
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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. 

29th August 2020, 10:12 AM  #286 
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29th August 2020, 11:02 AM  #287 
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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. 

29th August 2020, 11:11 AM  #288 
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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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29th August 2020, 11:15 AM  #289 
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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. 

29th August 2020, 11:19 AM  #290 
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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. 

29th August 2020, 11:30 AM  #291 
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29th August 2020, 11:34 AM  #292 
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29th August 2020, 11:44 AM  #293 
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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. 

29th August 2020, 12:26 PM  #294 
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Your most recent set of supporting definitions are these: 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: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 socalled definitions are little more than a distraction from the goal. 
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29th August 2020, 12:29 PM  #295 
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29th August 2020, 11:11 PM  #296 
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"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. 

30th August 2020, 12:01 AM  #297 
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Originally Posted by jsfisher

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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. 

30th August 2020, 12:09 AM  #298 
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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. 

30th August 2020, 07:43 AM  #299 
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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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30th August 2020, 08:39 AM  #300 
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30th August 2020, 08:45 AM  #301 
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30th August 2020, 08:50 AM  #302 
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31st August 2020, 04:46 AM  #303 
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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. 

31st August 2020, 05:01 AM  #304 
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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. 

31st August 2020, 06:18 AM  #305 
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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).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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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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31st August 2020, 07:34 AM  #307 
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OK, I see what you mean, for example:
{} ∪ any nonempty set is that nonempty 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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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. 

31st August 2020, 07:52 AM  #308 
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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. 

31st August 2020, 09:15 AM  #309 
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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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1st September 2020, 08:27 AM  #310 
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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 1: A set X is a successor set iff (x∪{x}).
Originally Posted by jsfisher
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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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. 

1st September 2020, 09:02 AM  #311 
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1st September 2020, 09:16 AM  #312 
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2nd September 2020, 01:18 AM  #313 
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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. 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. 

2nd September 2020, 01:40 AM  #314 
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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. 

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. 

2nd September 2020, 03:50 AM  #316 
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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 ... ... } 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 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 welldefine ∀ the members of set U (the term ∀ itself is actually not welldefined 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. 

2nd September 2020, 11:13 AM  #318 
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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:

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2nd September 2020, 11:02 PM  #319 
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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 ... ... } 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 welldefine ∀ the members of set U (the term ∀ itself is actually not welldefined 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. 

3rd September 2020, 02:39 AM  #320 
Penultimate Amazing
Join Date: Mar 2008
Posts: 13,260

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. 
__________________
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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