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4th September 2020, 02:09 PM  #361 
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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 2020, 02:10 PM  #362 
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4th September 2020, 02:13 PM  #363 
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4th September 2020, 02:17 PM  #364 
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4th September 2020, 02:19 PM  #365 
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Nothing has to be corrected in the following definitions:
Definition 1: A set K is a base set iff K ≠ {x : x∈x AND {x∈x}}. Definition 2: A set K is a successor set iff K = {x : x∈x AND {x∈x}}. As long as you "correct" them you are simply missing http://www.internationalskeptics.com...&postcount=348 . 
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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 2020, 02:25 PM  #366 
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4th September 2020, 07:56 PM  #367 
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Looking over the past many pages of this thread, it is clear, Doronshadmi, you are struggling. You are trying desperately to piece together snippets from a language you don't understand to express ideas you don't understand in opposition to concepts you don't understand.
At some point in the past a raised an analogy to Pakleds. I'll let others decide if the analogy has any present merit. Still, in the hopes of some minimal progress past the barrier that now stymies you, Doronshadmi, I offer this: Is this what you are trying to say for Definition 2? Definition 2: S is a successor set ⇔ ∃p∈S (S = p∪{p})(It has to be Comic Sans. It just has to be.) Possible responses:
Otherwise, please stop pretending. 
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5th September 2020, 12:36 PM  #368 
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Not if x is a placeholder for members, or for not members at all (for example, in the case of {}).
Since the notion of emptiness or nothingness is not directly formalized by ZF (ZF deals only with, so called, 'existing' objects, where the 'minimal existing' object is {}) {x∈x} or x∈x are indeed taken as gibberish, by the standard formalization. More details about this issue are already given in http://www.internationalskeptics.com...&postcount=352. 
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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. 

5th September 2020, 01:12 PM  #369 
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jsfisher, first "Yes, thank you; I understand" you have formalized successor set by the standard way without the need for ∀.
Second, please hold your horses of criticism about me, maybe you forgot that couple of days ago, you said that there is no formal definition for successor set. Moreover, now N members are formally defined by their structures as (base sets) OR (successor sets that have their successor sets) and we can move on in our discussion. 
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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. 

5th September 2020, 01:44 PM  #370 
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You're welcome. It was clear you were not going to ever get there. By the way, although I've expressed what I think you were trying to express, it still isn't what you meant. There are hidden assumptions you have yet to expose.
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5th September 2020, 01:45 PM  #371 
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5th September 2020, 11:36 PM  #372 
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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. 

6th September 2020, 06:08 AM  #373 
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Let's start from your first use of the terms base elements and their successors.
As can be seen you are using these terms in order to define set N, without first formally define a base element or its successors (which means that by this its there is an inseparable linkage between the two definitions, where X is a set by your own words (you are using it in your definition, after all)). By reducing N base elements into a single element (the empty set) you established N as the set of von Neumann ordinals, and then you defined injection from N to X, in order to formally define X as a nonfinite set (where N is a proper subset of X and the minimal nonfinite set established by the axiom of infinity). You did not formally define base element or its successors, and so is the case of set X, you did not define its membership function, yet you used it in your definition of X as a nonfinite set. When I struggled to define N as the minimal set, by trying to formally define base element (or base set), you "have changed the rules of the game" and imposed base set {{{}}} on it, as if N is actually X and not the minimal set (the set of von Neumann ordinals), as clearly seen in http://www.internationalskeptics.com...&postcount=202. By doing that you have created a confusion in our discussion, and I had to change N into X. When it was done I became aware of the beautiful verity of base sets and their successors as bijective proper subsets of set X (was called N by you from now on, as seen in the link above, where V became what was N (V became the set of von Neumann ordinals)). You ask me where you said that there is no formal definition for what I called successor set. Well, it is my mistake, you did not directly said that but indirectly, because of the linkage between the definitions of base set and a successor set. Here is the relevant quote, taken from http://www.internationalskeptics.com...&postcount=206 : Here is your definition of successor set: S is a successor set ⇔ ∃p∈S (S = p∪{p}) B is a base set ⇔ not a successor set. So as can be seen, now there is a formal definition of base set, which is something that you said that it has no formal definition, either . In other words, without my struggle to define formal definitions to successor set (which stands at the basis of the formal definition of base set) and base set as members of set N (which was first named by you as set X), we can't be at the current point in our discussion, which enables to formally define largest successor, and use it in order to define the yes/no membership function for set N (which was actually used by you as set X, in order to define X as a nonfinite set, without first define the membership function of X). Moreover, my nonstandard use of x as a placeholder for members, or no members at all (as seen in case of {}), may open a door for direct formalization of nothingness. Furthermore, by formally define successor set we naturally avoid Russell's paradox, since no set is its successor set, for example: N ≠ N∪{N}. No ad hoc axioms ( as done in ZF by establish the axiom of specification https://en.wikipedia.org/wiki/Axiom_..._specification ) are needed.  jsfisher, without being open to each other, even if we are doing mistakes, there is less chance that new notions are emerged in order to be formalized later. In my opinion, this is how the evolution of ideas works, by mutations. During the past 12 years I gradually tried to learn how to take it as a journey and not as a war between egos. I invite you to establish a paradise of base sets and their successor sets: Code:
N = { ∅, > {{∅}}, > ... < base sets {∅}, > {{∅}, {{∅}} }, > ... < successor sets {∅, {∅}}, > {{∅}, {{∅}}, {{∅}, {{∅}}}}, > ... < successor sets {∅, {∅}, {∅, {∅}}}, > {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, > ... < successor sets ... ... } 
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6th September 2020, 08:37 AM  #374 
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6th September 2020, 09:01 AM  #375 
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Are you completely oblivious to the fact that was in informal description of a particular set I had in mind? Informal, key word there. I invented the term, base element, for the sole purpose of conveying a concept. I had no intention of going any deeper than creating a mental image.
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Be that as it may, Now that you have acceptabletoyou definitions for base set and successor set, will you be proceeding to Definition #3, now?
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6th September 2020, 09:15 AM  #376 
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While waiting for Doronshadmi's new and improved Definition #3, it might be useful considering some implications of his first two definitions.
Quote:
Here are some examples of base sets: ∅, and {{∅}}, and {∅, {{∅}}}, and {∅, {∅}, {∅, {{∅}}}} Here are some examples of successor sets: {∅}, and {{∅}, {{∅}}}, and {∅, {∅}}. and {∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} 
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7th September 2020, 03:48 AM  #377 
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Here are your first terms about set N :
Originally Posted by jsfisher
I call such move "changing the rules of the game in the middle of the game". You actually meant to set X, but used set N instead, which caused a confusion in our discussion. Moreover, where is exactly the membership function of X that is used in your definition, which defines X as a nonfinite set? As long as you do not do that, X is no more than a letter in your definition of nonfinite set (and so is N, it is no more than a letter in your definition of nonfinite set). Here is your definition of nonfinite set:
Originally Posted by jsfisher
Originally Posted by jsfisher
Shell we also take it as an informal definition, in spite of the iff there ? 
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7th September 2020, 03:55 AM  #378 
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How exactly {∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} is a successor set by the following definition? : Definition: S is a successor set ⇔ ∃p∈S (S = p∪{p}) In order to define S={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} you first have to define p. So what is p if S={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} ? 
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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. 

7th September 2020, 05:50 AM  #379 
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7th September 2020, 05:54 AM  #380 
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7th September 2020, 06:26 AM  #381 
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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. 

7th September 2020, 06:35 AM  #382 
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7th September 2020, 07:27 AM  #383 
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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. 

7th September 2020, 08:11 AM  #384 
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Lining up sets to show there is a difference
Edit reasons: Purposely edited messages to remove extra topics, added [code] tag in a post to help line up the elements, and added letters below the brackets show the "level" of the brackets. 
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7th September 2020, 08:43 AM  #385 
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Ah! Back to progressing towards that deadend that's just a few steps away.
You are using "successor set" in two different ways  one as a set property; the other as a set function. The former usage you have defined; you'll need something else for the later. Note for later, too, that to apply this definition you'll need to specify both K and X. 
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7th September 2020, 08:56 AM  #386 
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Maybe a simpler way is to use, for example, colors.
S is a set which its members are (the members that are induced in p) AND (the members that are induced in {p} (which is actually p)). p = { ∅, {∅}, {∅, {{∅}}} } {p} = { { ∅, {∅}, {∅, {{∅}}} } } In that case S={ ∅, {∅}, {∅, {{∅}}} , { ∅, {∅}, {∅, {{∅}}} } } or S={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}} since order is irrelevant. 
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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. 

7th September 2020, 09:20 AM  #387 
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Let's do some diet:
Definition 3: A successor set K is a largest successor set iff ∃K ∈ X AND K∪{K} ∉ X For example: K={∅} K successor = {∅,{∅}} = {∅}∪{{∅}} = K∪{K} Now, ∃K ∈ X = {{∅}} = {K} AND {∅,{∅}} = {∅}∪{{∅}} = K∪{K} ∉ 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. 

7th September 2020, 09:48 AM  #388 
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Good! That didn't take very long at all. However, the stuff after the AND is wrong, but easily fixed. The existential quantifier is completely wrong, too, and must be removed. K, after all, is a given. You also need to indicate that being a largest successor set is with respect to some set. Perhaps you meant:
Definition 3: A successor set K is a largest successor set in X iff K ∈ X AND K∪{K} ∉ XYou might consider, too, this simplification. It is only so slightly different in what can be a largest successor set (i.e., base sets), but I think it is closer to what you really mean: Definition 3: Given X and K, K is a largest successor set in X iff K ∈ X AND K ∪ {K} ∉ XNow, you just need a membership function, M(x), so you can declare N = { x : M(x) }. Remember, too, any reliance on Definition #3 needs to identify two sets. ETA: Actually, it is probably better to leave the set membership condition on the right. I've edited my second definition offering accordingly. 
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7th September 2020, 09:58 AM  #389 
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Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})
Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p})) Definition 3: A successor set L is a largest successor set iff ∃L ∈ X AND L∪{L} ∉ X N = {x : (x=base set) ∨ (x=successor set ∧ ~largest successor set)}  No set is its successor set since for any given set S, S ≠ S∪{S}, or in other words, we avoid Russell's Paradox without any need for special axioms. 
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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. 

7th September 2020, 10:09 AM  #390 
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7th September 2020, 10:55 AM  #391 
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Thank you a lot jsfisher.
Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p}) Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p})) Definition 3: A successor set L is a largest successor set iff ∃L ∈ X AND L∪{L} ∉ X N = {x : (x=base set) ∨ (x=successor set ∧ ~largest successor set)}  No set is its successor set since for any given set S, S ≠ S∪{S}, or in other words, we avoid Russell's Paradox without any need for special axioms. 
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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. 

7th September 2020, 11:01 AM  #392 
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And yet you didn't correct any of the problems in Definition #3.
Quote:
Quote:

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7th September 2020, 11:08 AM  #393 
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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. 

7th September 2020, 11:20 AM  #394 
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Well, Ok let's correct it.
Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p}) Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p})) Definition 3: A successor set L is a largest successor set in X iff L ∈ X AND L∪{L} ∉ X N = {x : (x is a base set) ∨ (x is successor set ∧ ~largest successor set)} N can't be its own member by definition 1, so N is not the set of all sets. Here is some example of set N: Code:
N = { ∅, > {{∅}}, > ... < base sets {∅}, > {{∅}, {{∅}} }, > ... < successor sets {∅, {∅}}, > {{∅}, {{∅}}, {{∅}, {{∅}}}}, > ... < successor sets {∅, {∅}, {∅, {∅}}}, > {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, > ... < successor 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. 

7th September 2020, 11:32 AM  #395 
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Now about Russell's Paradox.
∀ 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. 

7th September 2020, 11:53 AM  #396 
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Please show where exactly I used N before you used it in http://www.internationalskeptics.com...5&postcount=83 ?

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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. 

7th September 2020, 12:03 PM  #397 
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7th September 2020, 12:11 PM  #398 
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7th September 2020, 12:16 PM  #399 
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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. 

7th September 2020, 12:39 PM  #400 
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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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