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19th August 2019, 04:23 AM  #3401 
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From my first post about A, A is a formal system of infinitely many wffs (which is strong enough in order to deal with Arithmetic, exactly because it is an extension of ZF(C)), where all the infinitely many wffs are already included in A, exactly because Infinity is taken in terms of Platonic (or Actual) Infinity (By Platonic (or Actual) Infinity there exists a set of infinitely many things (for example: wffs) as a complete whole).
The maneuvers of jsfisher around A's existence, this is exactly the thing that makes no sense. Here is jsfisher's last reply, which clearly demonstrates his nonsensical maneuvers around A's existence: and he does them in order to avoid the following question: Keep in mind that ZF(C) Axiom Of Infinity is taken in terms of Platonic (or Actual) Infinity, such that there a exists a certain set with infinitely many things as a complete whole. I take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A. Nothing more. Now, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? (To the other posters:
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Moreover
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Also, Infinity is one of the main philosophical subjects, studied by philosophers like Plato, Aristotle and many more philosophers along the years. The attepmt to define a clear cut distinction between Philosophy and Mathematics in case of Logic and Infinity, is itself some kind of Philosophy, and in this case jsfisher's philosophy about the discussed subject) 
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19th August 2019, 06:13 AM  #3402 
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Almost. The Axiom is not "taken"; it simply is. A certain set exists; it has certain properties. The von Neumann ordinal is the minimal example of such a set, so we have one example of the set guaranteed to exist. The Axiom alone gives no guidance as to whether there are others.
Your insistence on bringing in philosophic babble is, well, yours.
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You probably want to clean that up, and when you do, please explain what "relate it to" means. 
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19th August 2019, 07:05 AM  #3403 
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Let's see: "The Axiom is ... simply is."
jsfisher, maybe this is a very interesting statement. Probably a lot of mathematical work can by done by it and maybe also a profound communication between people can be done by it. Unfortunately, I do not find such tautology as very useful in our discussion. Such set is guaranteed to exist in terms of Platonic (or Actual Infinity) Infinity (which according to it there exists an infinite set as a complete whole). Your insistence to establish a clear cut border between Philosophy and Mathematics (and in the discussed case, Logic) is, well, your philosophy. It means that a certain property of x is also a property of y, and in the considered case Infinity is established in terms of Platonic Infinity, both on x and y. Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
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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. 

19th August 2019, 07:14 AM  #3404 
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19th August 2019, 07:44 AM  #3405 
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The certain property is Platonic (or Actual) Infinity, which is related to ZF(C) Axiom Of Infinity (x) and ZF(C) extension (y).
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ω is not established without the existence of the infinite set of all natural numbers, and the infinite set of all natural numbers is not established (by ZF(C) Axiom Of Infinity) as a complete whole (which enables ω to exist "After all natural numbers") without the "philosophic babble" of Platonic (or Actual) Infinity. Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
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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. 

19th August 2019, 09:01 AM  #3406 
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The property would be "infinite" not "infinity", but let's move on.
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And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A? (By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.) 
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20th August 2019, 02:44 AM  #3407 
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Let's not move on. The certain property is Infinity in terms of Platonic (or Actual) Infinity.
Platonic Infinity is what The Axiom of Infinity establishes, by using finitely many symbols, for example: The infinite set of von Neumann ordinals (https://en.wikipedia.org/wiki/Natura...umann_ordinals). Axiom schema (and therefore Infinity) are parts of ZF(C). As done by Godel First Incompleteness Theorem, but in terms of Platonic (or Actual) Infinity (which means that all wffs (whether they are axioms or theorems) are already included in this extension (called formal system A). Formal system A has infinitely many wffs in terms of Platonic (or Actual) Infinity (formal system A is taken as a complete whole). In the considered case Z' (which is a complete extension of Z in terms of Platonic (or Actual) Infinity) is indeed strong enough to deal with Arithmetic (as decidable in Z), but unlike Z, all its infinitely many wffs are already included in it (Z' is complete in terms of Platonic Infinity) as follows: Each wff is encoded by a Gödel number, where at least one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in Z', of G" (since G needs a proof, it is not an axiom but a theorem). Since all wffs are already in Z' and all Gödel numbers are already in Z' (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in Z', which contradicts G in Z', exactly because Z' is complete (in terms of Platonic Infinity) and therefore inconsistent exactly because Infinity is taken in terms of Platonic (or Actual) Infinity. Conclusion: Platonic (or Actual) Infinity is the cause of the contradiction (and therefore the inconstancy) of Z' (which is an extension of Z).  Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
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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. 

20th August 2019, 06:28 AM  #3408 
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Many bits of nonsense saved for a later time so as to not further defocus the current thread arc.
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Quote:
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For it to exist, you would need a membership function with respect to your set, A, for the Axiom Schema of Restricted Comprehension. You just need a function that determines whether x is a member of A. You don't have one. Merely speculating that a set must exist because you want it to does not make it so. Throwing your philosophic baggage at the problem doesn't change that. 
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20th August 2019, 08:01 AM  #3409 
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Another hands waving of yours.
I agree with you, G is just one statement, as written at the end of my previous post (" Z' " is used instead of "A"). In order to claim that there are countably infinite number of axioms, you first have to accept that there is an infinite set in terms of a complete whole. This is a finite GIT version that can't deduce anything about a set in terms of Platonic (or Actual) Infinity. By GIT infinite version (in terms of Platonic (or Actual) Infinity, which, as can be seen, was not deduced by you) all the Godel numbers that encode wffs, are already in A, where one of them encodes G wff statement (which is actually a theorem, since it is proven in A). The Axiom Schema of Restricted Comprehension does not exist, if Infinity is not taken in terms of Platonic (or Actual) Infinity. If you reject what I wrote about this axiom, you also reject the existence of the infinite set of all natural numbers as a complete whole. Your clear cut separation between Philosophy and Mathematics, does not change the fact that it is actually your Philosophy.  Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
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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. 

20th August 2019, 09:15 AM  #3410 
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You are making up your own theorems, now, too?
Quote:
The set theory corresponding to your set, A, is not an extension of ZF (or ZFC). There are statements in ZF that would be contradicted in your socalled extended set theory. As I said before, the set you claim exists does not. You require it represent an extension to ZF that is Godel complete. No such set exists. 
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20th August 2019, 10:55 PM  #3411 
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It is made up exactly as ZF(C) Axiom Of Infinity made up things in terms of Platonic (or Actual Infinity), which enables mathematicians like you to declare that the infinite set if all natural numbers, exists.
This is my argument right from the beginning of the last discussion, which is: The very notion of Platonic (or Actual) Infinity necessarily involved with logical contradiction and therefore inconsistency, exactly because a collection of infinitely many things is taken in terms of a complete whole. Until this very moment you are still missing my argument, which is (again, since you are still missing it): The very notion of Platonic (or Actual) Infinity necessarily involved with logical contradiction and therefore inconsistency, exactly because a collection of infinitely many things is taken in terms of a complete whole. Because of this logical fallacy also the infinite set of all natural numbers does not exist. Actually, the very notion of Transfinite System does not exist (in terms of logical consistency), exactly because it is established on the notion of collection of infinitely many things in terms of a complete whole.  Again, there is a noninteresting solution about the discussed subject, as follows: G states: "There is no number m such that m is the Godel number of a proof in A, of G" If G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A, which does not have any Godel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A). But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).  Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
__________________
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. 

21st August 2019, 04:14 AM  #3412 
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21st August 2019, 04:47 AM  #3413 
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Let's see:
"In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I." (Please compare it to https://en.wikipedia.org/wiki/Axiom_...rmal_statement, where m is replaced by x). If Infinity in this axiom is not taken in terms of Platonic (or Actual) Infinity, even the infinite set of all natural numbers does not exist and jsfisher's "...just mathematics" actually does not establish the Transfinite system.  jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
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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. 

23rd August 2019, 01:47 AM  #3414 
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Syntactically (by formalism without semantics) There is a set A (the set which is postulated to be infinite), such that the empty set is in A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x} is also a member of A.
So Syntactically x > xU{x} is the bijective membership function of A. Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows: Model 1: Let x be an axiom (wff that is not proven) in A. Let xU{x} be a theorem (wff that is proven) in A. Let A be an infinite set of wffs, where Infinity is taken in terms of Platonic (or Actual) Infinity (A is taken as a complete whole). Each wff (wff that is proven) is encoded by a Gödel number, where one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in A, of G" (since G needs a proof, it is not an axiom but a theorem). Since all wffs (wffs that are proven) are already in A and all Gödel numbers are already in A (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in A, which contradicts G in A, exactly because A is complete (in terms of Platonic Infinity) and therefore inconsistent. Model 2: Let x or xU{x} be axioms (wff that is not proven) in A. G axiom states: "There is no number m such that m is the Godel number of a proof in A, of G" Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A, which does not have any Godel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A). But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution). 
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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. 

23rd August 2019, 03:59 AM  #3415 
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So in both models infinitely in terms of Platonic (or Actual) Infinity (which according to it there exists a collection of things as a complete whole) does not establish an interesting formal system.
An alternative to such noninteresting formal systems, is established such that Platonic (or Actual) Infinity is noncomposed (it is not established in terms of collections that are taken as a complete whole) as already given in the following posts: http://www.internationalskeptics.com...postcount=3302 http://www.internationalskeptics.com...postcount=3303 http://www.internationalskeptics.com...postcount=3304 http://www.internationalskeptics.com...postcount=3305 
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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 2019, 06:30 AM  #3416 
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A clearer version of my argument (as firstly was given in http://www.internationalskeptics.com...postcount=3414)
Gödel numbers are used to encode wffs of formal systems that are strong enough in order to deal with Arithmetic. In my argument, Gödel numbers are used to encode wffs as follows: Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A. So Syntactically x → xU{x} is the bijective membership function of A. Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows: Model 1: Let any x be an axiom (wff that is not proven) in A. Let any xU{x} be a theorem (wff that is proven) in A. Let A be an infinite set of wffs, where Infinity is taken in terms of Actual Infinity (A is taken as a complete whole). Each wff (wff that is proven (some xU{x})) is encoded by a Gödel number, where one of these wffs, called G, states: "There is no number m such that m is the Gödel number of a proof in A, of G". Since all wffs are already in A and therefore all Gödel numbers are already in A (because Infinity is taken in terms of Actual Infinity) there is a Gödel number of an axiom (some x) that proves G (some xU{x}) in A, which is a contradiction in A. Therefore, A is inconsistent. Model 2: Let any x or any xU{x} be axioms (wffs that are not proven) in A. G axiom states: "There is no number m such that m is the Gödel number of a proof in A, of G" Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Actual Infinity) it is actually a wff that is true in A, which does not have any Gödel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A). But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).  So, in both models infinitely in terms of Actual Infinity (an infinite set that is taken as a complete whole) does not establish an interesting formal system.  Please pay attention to the following remarks, before you reply: Since A is a set of infinitely many wffs that are taken as a complete whole (this is exactly what Actual Infinity is about) there cannot be a Gödel number that is not already in A, whether whether some wff is an axiom or a theorem in A (see Model 1). So, one can't use G as a wff that is unproven in A, as done in case of GIT, since if one does this, one deduces in terms of Potential Infinity, which is not a part of my argumnt. 
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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 2019, 01:18 PM  #3417 
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3rd September 2019, 02:34 AM  #3418 
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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 2019, 04:39 PM  #3419 
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Repeating your false claim doesn't make it true. Nothing maps to the empty set.
Be that as it may, it isn't a membership function, either, since it doesn't answer the question, "Is m a member of set A?" F(x) = x U {x} doesn't.
Quote:
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. 
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3rd September 2019, 10:27 PM  #3420 
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Repeating your false claim doesn't make it true. As for the empty set, it is a member of A (as given below) and nothing maps to the empty set since it is a domain object (as seen in http://www.internationalskeptics.com...postcount=3418).
There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A. So, both x and xU{x} are members of A. So, Syntactically x → xU{x} is the bijective membership function of A (no x, xU{x} or any gödel number m are missing from A). m is a Gödel number of a proof of some xU{x} in A (according to model 1 (seen in http://www.internationalskeptics.com...postcount=3416)) exactly because A is determined in terms of Actual Infinity as a complete whole (no m, x or xU{x} are missing from A). I am not talking about ZF, but about the construction of A such that Infinity is taken as a complete whole (no x, xU{x} or any gödel number m are missing from A). If you don't like xU{x}, it can easily be replaced by {x} as follows: {} → {{}} {{}} → {{{}}} {{{}}} → {{{{}}}}} ... etc. For example, syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set {x}, is also a member of A. So, both x and {x} are members of A. So, Syntactically x → {x} is the bijective membership function of A (no x, {x} or any gödel number m are missing from A). 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

4th September 2019, 05:17 AM  #3421 
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4th September 2019, 07:27 AM  #3422 
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The two following equivalent examples are taken from Traditional Mathematics:
Example 1: There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A. So, both x and xU{x} are members of A. {} → {{}} {{}} → {{},{{}}} {{},{{}}} → {{},{{}},{{},{{}}}} ... etc. are all members of A in example 1. So, Syntactically x → xU{x} is the bijective membership function of A (no x, xU{x} or any gödel number m are missing from A). Example 2: There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by {x}, is also a member of A. So, both x and {x} are members of A. {} → {{}} {{}} → {{{}}} {{{}}} → {{{{}}}}} ... etc. are all members of A in example 2. So, Syntactically x → {x} is the bijective membership function of A (no x, {x} or any gödel number m are missing from A).  Example 2 easily replaces example 1 (which is used in http://www.internationalskeptics.com...postcount=3416) without changing the argument.  Wrong , {} → {{}} and {{}} → {} are inverses of each other.  jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)? 
__________________
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. 

7th September 2019, 11:28 AM  #3423 
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Please observe the following semantic diagram:
As can be seen, there is bijection between XAxioms and YTheorems, as follows: {} → {{}} {{}} → {{},{{}}} {{},{{}}} → {{},{{}},{{},{{}}}} ... → ... It is a bijection since XAxioms and YTheorems are inverses of each other as follows: Xaxiom proves Ytheorem and Ytheorem is proven by Xaxiom. 
__________________
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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