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 International Skeptics Forum Continuation Deeper than primes - Continuation 1/3*9

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 20th July 2020, 09:23 AM #81 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi My point is straightforward: Traditional mathematics does not consistently and rigorously establish a non-finite set. Great! Now, how do you support this point of view? __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 20th July 2020, 11:56 PM #82 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher Great! Now, how do you support this point of view? 1) There is nothing in your definition of relative cardinality (|A| <= |B| if and only if there exists an injection from A to B) which establishes a non-finite set. 2) The ZF(C) axiom of infinity establishes the existence of (at least one) set, I that has those two properties: Property 1: ∅ ∈ I Proeprty 2: ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) This axiom by itself does not establish a non-finite set, no matter what name is given to it. 3) No axiom schema can help here since it is by itself a set of axioms that can't be non-finite as long as a non-finite set is not consistently and rigorously established. 4) You can use any relations that you like among ZF axioms, but they can't establish a non-finite set in a way that is not arbitrary, exactly because cardinality is not directly defined by it. Originally Posted by jsfisher Subtlety and nuance escape you. I did not define cardinality, at least not directly. Euclidean Geometry does not define point (directly), nor does ZF Set Theory define set (directly). Instead, ZF Set Theory provides a set of axioms and axiom schema that characterize set properties; Euclidean Geometry characterizes points by way of its postulates; and I defined a relationship sufficient for comparing cardinalities of sets. Guess what that does. I do not have to guess, as long as your system does not define Cardinality directly, you can't consistently and rigorously establish a non-finite set. __________________ 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; 21st July 2020 at 12:31 AM.
 21st July 2020, 04:11 AM #83 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi 1) There is nothing in your definition of relative cardinality (|A| <= |B| if and only if there exists an injection from A to B) which establishes a non-finite set. Yes, very true. My approach to cardinality applies to all sets, not just some. Quote: 2) The ZF(C) axiom of infinity establishes the existence of (at least one) set, I that has those two properties: ... This axiom by itself does not establish a non-finite set, no matter what name is given to it. Another true statement. Then again, what's missing from all this? What is it that you, Doronshadmi, continually overlook because you continue to apply an elementary school perspective to all of Mathematics? Answer: A meaning for 'non-finite set'. And how could that be remedied? A simple definition, perhaps, of what it means for a set to be non-finite? Maybe something like:A set, X, is non-finite if and only if |N| <= |X|, where N is the von Neumann ordinal.Other definitions are possible, but this one will do. Quote: I do not have to guess, as long as your system does not define Cardinality directly, you can't consistently and rigorously establish a non-finite set. And yet reality would disagree with you. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 21st July 2020, 04:43 AM #84 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher A simple definition, perhaps, of what it means for a set to be non-finite? Maybe something like:A set, X, is non-finite if and only if |N| <= |X|, where N is the von Neumann ordinal.Other definitions are possible, but this one will do. |N| <= |X| can be written as (|N|<|X|) OR (|N|=|X|). In that case you consistently and rigorously need to show that N is a non-finite set (after all we discuss about mathematics, isn't it?) Please provide another definition or consistently and rigorously show that N is a non-finite set. Originally Posted by jsfisher And yet reality would disagree with you. Please define reality. __________________ 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; 21st July 2020 at 04:47 AM.
 21st July 2020, 04:49 AM #85 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi |N| <= |X| can be written as (|N|<|X|) OR (|N|=|X|). Other than your insistence on adding unnecessary things, why would we need to do that? Quote: In that case you constantly and rigorously need to show that N is a non-finite set. Since |N| <= |N|, done! __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 21st July 2020, 04:59 AM #86 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher Since |N| <= |N|, done! Not done as long as you consistently and rigorously show that N is a non-finite set (after all we discuss about mathematics, isn't it?) Please provide another definition or consistently and rigorously show that N is a non-finite set. __________________ 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 July 2020, 05:05 AM #87 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi Not done as long as you consistently and rigorously show that N is a non-finite set (after all we discuss about mathematics, isn't it?) You don't understand how definitions work, do you? The term, non-finite set, has been defined. N is non-finite by that defnition. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 21st July 2020, 05:29 AM #88 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 A Wikipedia quote (https://en.wikipedia.org/wiki/Ordina...on_of_ordinals): Quote: First few von Neumann ordinals Code: ```0 = { } = ∅ 1 = { 0 } = {∅} 2 = { 0, 1 } = { ∅, {∅} } 3 = { 0, 1, 2 } = { ∅, {∅} , {∅, {∅}} } 4 = { 0, 1, 2, 3 } = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ...``` A Wikipedia quote: Quote: Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω,... In this case one first has to consistently and rigorously show that the non-finite set of all natural numbers, exists. __________________ 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 July 2020, 05:31 AM #89 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher You don't understand how definitions work, do you? The term, non-finite set, has been defined. N is non-finite by that defnition. Sorry but your link does not work. If you mean to "A set, X, is non-finite if and only if |N| <= |X|, where N is the von Neumann ordinal.", then, as very simply shown, this definition does not satisfy http://www.internationalskeptics.com...9&postcount=88. Your "definition" is a good example of how mathematics is arbitrarily done. __________________ 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; 21st July 2020 at 05:44 AM.
 21st July 2020, 08:15 AM #90 Little 10 Toes Master Poster     Join Date: Nov 2006 Posts: 2,230 Please define "non-finite set". Because you do not use the term "infinite set", your term must have a special meaning to you. __________________ I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." -Tim4848 who said he would no longer post here, twice in fact, but he did.
 21st July 2020, 08:43 AM #91 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi In this case one first has to consistently and rigorously show that the non-finite set of all natural numbers, exists. No, one doesn't. One would need to show only that the von Neumann ordinal exists within set theory, which it does, but I'll leave that bit of trivia to you to show. (Hint: Axioms of Infinity and of Restricted Comprehension.) The definition I have provided for "non-finite set" is sufficient to show the von Neumann ordinal as a non-finite set. If you want to label the members of the von Neumann ordinal with natural numbers, go right ahead, but that's your doing, not a requirement of mine. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 21st July 2020, 08:58 AM #92 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 I apologize. It seems I have introduced some confusion I did not intend by using the term "von Neumann ordinal" in a non-standard way. The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...). A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element. So, a set X is a non-finite set if and only if |N| <= |X|, where N is the minimal set established by the Axiom of Infinity. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 22nd July 2020, 02:35 AM #93 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher So, a set X is a non-finite set if and only if |N| <= |X|, where N is the minimal set established by the Axiom of Infinity. ZF Axiom of Infinity is a good example of how Traditional Mathematics arbitrarily forces N to actually be non-finite by simply ignoring the fact that the largest successor does not exist as one of N's members. As a result |N| (N size) can't be defined since N is closed under this impossible existence. So to say that "a set X is a non-finite set if and only if |N| <= |X|" is simply nonsense. Here is how this nonsense is done: Originally Posted by jsfisher The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...). ...and their successors, and their successors, and ..., which is a fundamental property that can't able The Axiom of Infinity to establish N. Since N is not actually established, anything that is based on it, is not established. It has to be clear that non-finite collection of members is taken in terms of actual infinity by traditional mathematicians, in order to establish N. --------------------- The claim that |N| exists as an extension of N members, without directly define Cardinality, has no consistent and rigor foundations. __________________ 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; 22nd July 2020 at 03:49 AM.
 22nd July 2020, 02:56 AM #94 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by Little 10 Toes Please define "non-finite set". Because you do not use the term "infinite set", your term must have a special meaning to you. Please read very carefully http://www.internationalskeptics.com...9&postcount=73 including its link. __________________ 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.
 22nd July 2020, 04:48 AM #95 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi ZF Axiom of Infinity is a good example of how Traditional Mathematics arbitrarily forces N to actually be non-finite by simply ignoring the fact that the largest successor does not exist as one of N's members. As a result |N| (N size) can't be defined since N is closed under this impossible existence. You are arguing in circles. You cannot disprove definitions. And you certainly cannot arbitrarily apply your own definitions to do so. But you did. So, how do you define what it means for a set to be non-finite. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 22nd July 2020, 04:53 AM #96 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi Please read very carefully http://www.internationalskeptics.com...9&postcount=73 including its link. Nothing was defined there. Quite the opposite. There you insisted you didn't need to define anything because it was intuitive. This cycles back to your infamous proof by direct perception -- if you, Doronshadmi and only you, really, really believe something in Mathematics to be true, well, then, it must be true, because that's what you were taught on or about the third grade. That's a great self-serving philosophy, I suppose, but not particularly useful nor convincing. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost Last edited by jsfisher; 22nd July 2020 at 04:56 AM.
 22nd July 2020, 05:50 AM #97 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher Nothing was defined there. Because you ignore the link there. __________________ 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.
 22nd July 2020, 06:44 AM #98 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Here is an improved version of my definitions (which, unlike you, I do not take a definition as something that can't be changed by better notions). Let A be any arbitrary set. Let N be the set of natural numbers. Definition 1: Cardinality is the "size" of A (notated as |A|) iff it is related to the members of N, where 0 is a member of N. Let < be less than. Let > be more than. Let = be equal to. A relative measure of "sizes" of two sets does not have to be done by specific directions (the terms "from A to B" or "from B to A" are irrelevant). Definition 2: The cardinality between the two sets A and B is a relative measure of their "sizes", where ( ((|A| < |B|) iff (|B| > |A|)) OR ((|A| = |B|) iff (|B| = |A|)) ). (visually in "((|A| < |B|) iff (|B| > |A|))" there are two different symbols ("<" and ">"), where in "((|A| = |B|) iff (|B| = |A|))" there is the same symbol ("="), which intuitively reinforces the meaning of inequality or equality). These definitions say nothing about being finite or non-finite sets. So: Originally Posted by jsfisher You are arguing in circles. how do you define what it means for a set to be non-finite. Definition 3: Set A is called finite iff given any n in N, |A| is any particular n Definition 4: Set A is called non-finite iff given any n in N, |A| is not any particular n -------------------------------------------- By the standard notion "given any" is the same as "for all" ( as seen in https://en.wikipedia.org/wiki/Universal_quantification ) but not in my framework, where "given any" holds for both finite and non-finite sets, where "for all" holds only for finite sets. Let ≤ be less than or equal to. Let ≥ be greater than or equal to. Definition 5: The cardinality between sets A and B is called non-strict inequality iff ((|A| ≤ |B|) iff (|B| ≥ |A|)). __________________ 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; 22nd July 2020 at 06:51 AM.
 22nd July 2020, 10:10 AM #99 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi Let N be the set of natural numbers. Definition 1: Cardinality is the "size" of A (notated as |A|) iff it is related to the members of N, where 0 is a member of N. I doubt that is what you meant. "If and only if" is a boolean relationship with operands that are either true or false. When used as part of a definition, the if-and-only-if is taken as a constraint requiring the left-hand side (the thing being defined) bear the same truth value as the right-hand side. You are claiming that the left-had side, "Cardinality is the 'size' of A", is a statement that can be either true or false. On the other hand, the right-hand side, "[set A] is related to the members of N", appears to be a true/false valued proposition as would be required, but what does "related to" mean? Your claimed Definition 1 isn't a definition, and it introduces another undefined term. What is your definition for "non-finite set"? __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 23rd July 2020, 12:31 AM #100 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher I doubt that is what you meant. "If and only if" is a boolean relationship with operands that are either true or false. When used as part of a definition, the if-and-only-if is taken as a constraint requiring the left-hand side (the thing being defined) bear the same truth value as the right-hand side. You are claiming that the left-had side, "Cardinality is the 'size' of A", is a statement that can be either true or false. On the other hand, the right-hand side, "[set A] is related to the members of N", appears to be a true/false valued proposition as would be required, but what does "related to" mean? Your claimed Definition 1 isn't a definition, and it introduces another undefined term. What is your definition for "non-finite set"? Thank you. Being a member of N is not determined only by some specific property of that member, where "related to" is general enough in order to capture both the common or the specific properties of being a member of N. For example: Not any given N member = 0 (common property), but any given element = 0 is N member (specific property). __________________ 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; 23rd July 2020 at 02:23 AM.
 23rd July 2020, 02:39 AM #101 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher "[set A] is related to the members of N" No, By definition 1 the cardinality of A is related to the members of N, whether some particular n=|A|, or |A| is not any particular member of N. |A| is not any particular member of N exactly because of the impossibility to define N's largest successor (http://www.internationalskeptics.com...8&postcount=93). Please carefully read again http://www.internationalskeptics.com...7&postcount=98. Thank you. __________________ 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; 23rd July 2020 at 03:54 AM.
 23rd July 2020, 04:13 AM #102 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi No, By definition 1 the cardinality of A is related to the members of N What does "is related to" mean? Quote: ...whether some particular n=|A|, or |A| is not any particular member of N. For any give set A, which particular n (or not any) would that be? __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 23rd July 2020, 04:50 AM #103 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher What does "is related to" mean? http://www.internationalskeptics.com...&postcount=100 Originally Posted by jsfisher For any give set A, which particular n (or not any) would that be? Please look at definitions 3 and 4 in http://www.internationalskeptics.com...7&postcount=98. __________________ 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 July 2020, 06:19 AM #104 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi http://www.internationalskeptics.com...&postcount=100 Please look at definitions 3 and 4 in http://www.internationalskeptics.com...7&postcount=98. I have. There were a lot of words in those posts, but nowhere did you define anything. How, for example, would one determine that |D| = 4 and not 7 for some set D? __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 23rd July 2020, 07:16 AM #105 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 I wish to correct my example in http://www.internationalskeptics.com...&postcount=100, since it does not satisfy what I actually wish to say. A corrected example: N members have at least two common properties, which are quantity and/or order of the members of a given set (including N's members.) One of the ways to construct N members in terms of sets is done by "von Neumann ordinals" (http://www.internationalskeptics.com...9&postcount=88) and it is equivalent to ZF axiom of infinity. Originally Posted by jsfisher The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...). A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element. N members are constructed by no members (the empty set) or by all their finitely many predecessors so, given any arbitrary N successor it is a finite set whether it is defined in terms of "von Neumann ordinals" or its equivalent expression, called ZF axiom of infinity. So, no matter what names ("von Neumann ordinals" or "ZF axiom of infinity", etc.) are related to N, it has finitely many members. The only way to provide an illusion that N has infinitely many members is to define the cardinality of any arbitrary set, such that it is defined by N members, but not by any particular N member. By doing so, we ignore any particular N member and focused only on the fact that no N successor is the largest possible construction. So we have a starting point (the empty set) but no end point (because no N successor is the largest possible construction). The lack of an end point does not actually guarantee infinitely many N members, therefore N is at best can be defined in terms of potential infinity, and it is done by an arbitrary set A cardinality (where one of these sets is N itself) by N members, but not by any particular N member. As a result A finite cardinality is not satisfied, but it does not mean that "it opens the door" to actually infinite cardinality. So being non-finite is not actually be infinite, as give by the following definition: Definition 4: Set A is called non-finite iff given any n in N, |A| is not any particular n Originally Posted by jsfisher For any give set A, which particular n (or not any) would that be? Please look at definitions 3 and 4 in http://www.internationalskeptics.com...7&postcount=98. Originally Posted by jsfisher I have. There were a lot of words in those posts, but nowhere did you define anything. How, for example, would one determine that |D| = 4 and not 7 for some set D? By definition 3. Definition 3: Set D is called finite iff given any n in N, |D| is any particular n In means that |D| is defined by a particular n. __________________ 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; 23rd July 2020 at 07:31 AM.
 23rd July 2020, 07:35 AM #106 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 By going beyond the notion of collections (where sets are particular forms of collections) actual infinity is essentially non-composed, as already demonstrated in http://www.internationalskeptics.com...postcount=3285. __________________ 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; 23rd July 2020 at 07:37 AM.
 23rd July 2020, 09:52 AM #107 zooterkin Nitpicking dilettanteAdministrator     Join Date: Mar 2007 Location: Berkshire, mostly Posts: 46,722 Originally Posted by doronshadmi By definition 3. Definition 3: Set D is called finite iff given any n in N, |D| is any particular n In means that |D| is defined by a particular n. How do you know which particular n? __________________ The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.Bertrand Russell Zooterkin is correct Darat Nerd! Hokulele Join the JREF Folders ! Team 13232 Ezekiel 23:20
 23rd July 2020, 11:22 AM #108 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi A corrected example: N members have at least two common properties, which are quantity and/or order of the members of a given set (including N's members.) How do you define "quantity" as you are using the term? How will you represent "order" within what is provided within an axiomatic set theory framework? That's two things you need to define or explain vs. zero things you did define or explain...all within the first real sentence of your post. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 24th July 2020, 02:51 AM #109 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by zooterkin How do you know which particular n? Definition 3 simply states that D is finite iff |D| is a particular n. Definition 4 simply states that D is non-finite iff |D| is not a particular n. Please read http://www.internationalskeptics.com...&postcount=105 in order to understand the terms "a particular n" or " not a particular n", such that both terms are related to N members. __________________ 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; 24th July 2020 at 02:55 AM.
 24th July 2020, 05:49 AM #110 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher How do you define "quantity" as you are using the term? How will you represent "order" within what is provided within an axiomatic set theory framework? Quantity in this context is simply the number of members of a given set. Order in this context is simply the tags "first", "second", "third" etc. that we provide to the members of a given set. The empty set has zero members, where nothing can be ordered. __________________ 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; 24th July 2020 at 05:51 AM.
 24th July 2020, 10:06 AM #111 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi Quantity in this context is simply the number of members of a given set. How do you determine the number of members of a given set? Quote: Order in this context is simply the tags "first", "second", "third" etc. that we provide to the members of a given set. How are those assignments determined? How are those assignments represented within the bounds of an axiomatic set theory? __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost Last edited by jsfisher; 24th July 2020 at 10:30 AM.
 24th July 2020, 10:14 AM #112 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi Originally Posted by zooterkin How do you know which particular n? Definition 3 simply states that D is finite iff |D| is a particular n. Definition 4 simply states that D is non-finite iff |D| is not a particular n. Please read http://www.internationalskeptics.com...&postcount=105 in order to understand the terms "a particular n" or " not a particular n", such that both terms are related to N members. Please, answer zooterkin's question. Repeating things that do not answer the question isn't at all productive. You have multiple times stated something like "...if |D| is a particular n..." So, (1) how can you determine if |D| is a particular n or not, and (2) how would you determine which particular n it would be? (Both parts are implicit in zooterkin's single question.) __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 26th July 2020, 06:59 AM #113 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 zooterkin and jsfisher, please read all of http://www.internationalskeptics.com...&postcount=105. A given set D is finite if |D| is some particular N member. In case of a finite set, please think about |D| as a buss that stops at some given station n. In case of a non-finite set, please think about |D| as a buss that does not stop at any station, therefore |D| is not any particular n. Since N does not have the largest successor, |D| journey is endless, but can't cross into a given value > any given N member, unless it is done without any consistent and rigorous reasoning, for example: Originally Posted by jsfisher Since |N| <= |N|, done! You did not consistently and rigorously establish |N| as some value > any given N member, so you can write "done!" as much as you like, ZF axiom of Infinity or its equivalent expression known as "von Neumann ordinals" can't establish |N| as some value > any given N member, unless it is done whimsically. __________________ 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; 26th July 2020 at 07:36 AM.
 26th July 2020, 07:34 AM #114 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi zooterkin and jsfisher, please read all of http://www.internationalskeptics.com...&postcount=105. A given set D is finite if |D| is some particular N member. Given a set D that is finite, which particular member of N would that be? Quote: In case of a finite set, please think about |D| as a buss that stops at some given station n. (The word is 'bus'.) Which given station would that be? Quote: In case of a non-finite set, please think about |D| as a buss that does not stop at any station, therefore |D| is not any particular n. Given a set D, how is it determined whether it is finite or not? Had you actually defined cardinality and what it means to be a finite set, then these questions would not need to be asked, but you defined nothing. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 27th July 2020, 02:45 AM #115 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher Given a set D that is finite, which particular member of N would that be? (The word is 'bus'.) Which given station would that be? Given a set D, how is it determined whether it is finite or not? Had you actually defined cardinality and what it means to be a finite set, then these questions would not need to be asked, but you defined nothing. If the bus stops, then |D| is some n, and in this case D is a finite set (all we care is that the bus stops, in order to define D as a finite set, no matter what n=|D|). If the bus does not stop, then |D| is not any n, and in this case D is a non-finite set (all we care is that the bus does not stop, and in this case D is non-finite since |D| value can't be determined by some n and we also know that N dose not have the largest successor, so we are in an endless journey along N members and only along them. What is important here is "the bus stops" (D is finite) OR "the bus does not stop" (D is non-finite). So zooterkin question is irrelevant, in this case. You may say that (D is finite) OR (D is non-finite) is a tautology, but this tautology is found also by traditional mathematics. But traditional mathematics define |N| as > any given n without any basis. __________________ 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; 27th July 2020 at 04:38 AM.
 27th July 2020, 04:39 AM #116 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi If the bus stops, then |D| is some n.... How do you decide if it will stop? If it stops, where does it stop? __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 27th July 2020, 04:54 AM #117 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher How do you decide if it will stop? If it stops, where does it stop? Again, all we care is this (instead of "stops" we use "stands", and instead of "does not stop" we use "does not stand"): (the bus stands at some station (a given set is finite (the number of its members is determined by some n))) OR (the bus does not stand at any station) (a given set is non-finite (the number of its members is not determined by any n))) In both cases only N members are considered, and in the second case also the fact that N does not have the greatest successor, is considered. __________________ 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; 27th July 2020 at 05:10 AM.
 27th July 2020, 05:18 AM #118 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,927 Originally Posted by doronshadmi Again, all we care is this (instead of "stops" we use "stands", and instead of "does not stop" we use "does not stand"): I knew we were overdue for useless word substitutions. Quote: (the bus stands at some station (a given set is finite (the number of its members is determined by some n))) OR (the bus does not stand at any station) (a given set is non-finite (the number of its members is not determined by any n))) So how do you determine which one it does? If you don't provide a way to decide that question, then you haven't distinguished finite from non-finite. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 27th July 2020, 05:54 AM #119 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher So how do you determine which one it does? If you don't provide a way to decide that question, then you haven't distinguished finite from non-finite. EDITED: The bus stands at some station (no matter what station) means D is finite. The bus does not stand at any station means D is non-finite. That's all we need in order to distinguish between the finite and the non-finite. No decision of involved in order to distinguish between stands or does not stand. __________________ 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; 27th July 2020 at 06:06 AM.
 27th July 2020, 06:03 AM #120 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,250 Originally Posted by jsfisher I knew we were overdue for useless word substitutions. They are simply different words that are able to express the same notion. __________________ 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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