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

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 19th September 2020, 01:01 PM #521 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,920 Are you unable to tell us what you mean? Use your words. Examples are not definitions. Dodges are not definitions. __________________ 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
 19th September 2020, 03:12 PM #523 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 Originally Posted by jsfisher There is no set of the von Neumann ordinals simply because it is too big to be a set. V = { v : (v is not non-finite member of von Neumann ordinals) } = The set of all natural numbers in terms of sets. Originally Posted by jsfisher This is not a set description within the rules of ZF or ZFC. In that case the set of all natural numbers in terms of sets, is undefined within the rules of ZF or ZFC. Originally Posted by jsfisher And von Neumann's treatment of the natural numbers is right in line with my point: Just about everything in Mathematics is rooted in Set Theory. In other words, you are dishonest with yourself. |V| is not the cardinality of V simply because ∀v ∈ V(v∪{v} ∈ V ∧ |v∪{v}| < |V|). In simple words: For all v∪{v} in V, |v∪{v}| is too small in order to be valued as |V|. So, exactly as the von Neumann set of ordinals it is too big to be a set, so is the case of all natural numbers in terms of sets, it is too small to be a set. No wonder that λ was invented out of nowhere (as seen in http://www.internationalskeptics.com...&postcount=451) in order to cover that for all v∪{v} in V, |v∪{v}| is too small in order to be valued as |V|. Without the ad hoc invention out of nowhere of |V|=λ as a weak limit cardinal (such that λ is neither a successor cardinal nor zero) ZF(C) does not hold water. __________________ 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; 19th September 2020 at 03:47 PM.
 19th September 2020, 04:10 PM #524 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,920 Originally Posted by doronshadmi V = { v : (v is not non-finite member of von Neumann ordinals) } = The set of all natural numbers in terms of sets. This is not a valid definition for your set, V. It violates the Axiom of Restricted Comprehension, for example. Quote: In that case the set of all natural numbers in terms of sets, is undefined within the rules of ZF or ZFC. That does not follow. Just because you are unable to express, within the bounds of set theory, a definition of the minimal set that satisfies the Axiom of Infinity does not mean it cannot be done. From there, the natural numbers are not hard to reach. Getting to all the ordinals that von Neumann had in mind, as you insist on doing, is an extremely difficult and long route. It is also working backwards, as you are prone to do. And it also requires accepting as given all the things you are trying to dismiss. Why are you still dodging defining cardinality as you use the term? __________________ 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
 Yesterday, 06:29 AM #525 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 Originally Posted by jsfisher This is not a valid definition for your set, V. It violates the Axiom of Restricted Comprehension, for example. It is valid because of the following reasons: 1) All v in V and all v∪{v} in V are not non-finite sets, in terms of "von Neumann's treatment of the natural numbers". 2) For all v in V and for all v∪{v} in V, |v| is a strict cardinality of v, and |v∪{v}| is strict cardinality of v∪{v}. 3) V does not have strict cardinality since for all v∪{v} in V, |v∪{v}| < |V|, which means that |V| can't be a measurement value of V. 4) Without strict cardinality, V is not a set in terms of ZF(C), and therefore can't be a member of itself, in the first place. __________________ 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; Yesterday at 06:58 AM.
 Yesterday, 07:01 AM #526 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,920 Originally Posted by doronshadmi It is valid because of the following reasons... No part of your bogus argument matters until you define your terms. Cardinality is the big one you continue to dodge. Please define what you mean by cardinality. ETA: You will need to define "strict cardinality", too. __________________ 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; Yesterday at 07:06 AM.
 Yesterday, 01:45 PM #527 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 Originally Posted by jsfisher You can, in fact, use those meanings to show that the expression |A| <= |B| is identical to (|A| < |B|) OR (|A| = |B|) Definition 1: |A| = |B| iff there is bijection from A to B (where in case of bijection, B can be replaced by A and we get bijection from A to itself). V = { v : (v is not non-finite member of von Neumann ordinals) } = The set of all natural numbers in terms of sets. v or v∪{v} are not non-finite sets, where the cardinality of v (notated as |v|) or the cardinality of v∪{v} (notated as |v∪{v}|) is defined (by definition 1) by bijection form a given domain to itself. Definition 2: |A| < |B| iff (there is injection no surjection from A to B) OR (A=∅ ∧ B~=∅). ∀v ∈ V(v∪{v} ∈ V ∧ |v| < |v∪{v}|) by definition 2, which means that the expression |A| <= |B| (injection from A to B) is not satisfied from v to v∪{v} in V. Now let's look if |A| <= |B| (injection from A to B) can be used in order to establish |V|. ∀v ∈ V(v∪{v} ∈ V ∧ |v∪{v}| < |V|) In simple words: For all v∪{v} in V, |v∪{v}| is too small in order to be valued as |V|, which means that |V| can't be defined as the cardinality of all V members. In other words: |A| <= |B| (and definitely |V| <= |V|) do not hold water. As for the set of all natural numbers, |N| is not established. __________________ 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; Yesterday at 01:54 PM.
 Yesterday, 01:52 PM #528 Little 10 Toes Master Poster     Join Date: Nov 2006 Posts: 2,230 Originally Posted by doronshadmi Definition 1: |A| = |B| iff there is bijection from A to B (where in case of bijection, B can be replaced by A and we get bijection from A to itself). V = { v : (v is not non-finite member of von Neumann ordinals) } = The set of all natural numbers in terms of sets. v or v∪{v} are not non-finite sets, where the cardinality of v (notated as |v|) or the cardinality of v∪{v} (notated as |v∪{v}|) is defined (by definition 1) by bijection form a given domain to itself. Definition 2: |A| < |B| iff (there is injection no surjection from A to B) OR (A=∅ ∧ B~=∅). ∀v ∈ V(v∪{v} ∈ V ∧ |v| < |v∪{v}|) by definition 2, which means that the expression |A| <= |B| (injection from A to B) is not satisfied from v to v∪{v} in V. Now let's look if |A| <= |B| (injection from A to B) can be used in order to establish |V|. ∀v ∈ V(v∪{v} ∈ V ∧ |v∪{v}| < |V|) In simple words: For all v∪{v} in V, |v∪{v}| is too small in order to be valued as |V|, which means that |V| can't be defined as the cardinality of all V members. In other words: |A| <= |B| (and definitely |V| <= |V|) do not hold water. As for the set of all natural numbers, |N| is not established. doronshadmi can't even get bijection right. __________________ 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. Last edited by Little 10 Toes; Yesterday at 01:55 PM. Reason: added original post before doronshadmi starts editing it. again.
 Yesterday, 02:29 PM #529 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,920 Originally Posted by doronshadmi Definition 1: |A| = |B| iff there is bijection from A to B Great! except your definition doesn't exclude anything that would support your definition for non-finite set. 'Tis a problem, no? Quote: (where in case of bijection, B can be replaced by A and we get bijection from A to itself) You have a compulsion to overstate the obvious and irrelevant, don't you? Quote: V = { v : (v is not non-finite member of von Neumann ordinals) } = The set of all natural numbers in terms of sets. Not a valid definition. Plus, your "non-finite member" is in jeopardy of being an undefined term. Quote: Definition 2: |A| < |B| iff (there is injection no surjection from A to B) OR (A=∅ ∧ B~=∅). Ok, then. Still no support for your "non-finite set" definition. You'll need to try again defining that term. Kinda funny, too, you end up with substantially the same definition for cardinality as I, just with more steps. So, what is your definition for non-finite set? You broke the one you provided before. __________________ 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
 Yesterday, 03:04 PM #530 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,920 Doronshadmi, exactly what post of mine are you quoting in the following? Originally Posted by doronshadmi Originally Posted by jsfisher You can, in fact, use those meanings to show that the expression |A| <= |B| is identical to (|A| < |B|) OR (|A| = |B|) __________________ 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
 Yesterday, 10:40 PM #531 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 Originally Posted by jsfisher Doronshadmi, exactly what post of mine are you quoting in the following? http://www.internationalskeptics.com...97&postcount=2 __________________ 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.
 Today, 12:48 AM #532 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 Originally Posted by jsfisher Not a valid definition. Plus, your "non-finite member" is in jeopardy of being an undefined term. I start by using your definition of non-finite set, here it is: Originally Posted by jsfisher That would be back when I defined a set X to be non-finite when |N| <= |X| (where N is the minimal set satisfying the requirements of the Axiom of Infinity). Since finite and non-finite are complementary terms, defining one automatically defines the other So not non-finite set is a finite set by your terms. Originally Posted by jsfisher And von Neumann's treatment of the natural numbers is right in line with my point: Just about everything in Mathematics is rooted in Set Theory. Great, so by using von Neumann's treatment of the natural numbers, we define V as follows: V = { v : (v is not non-finite member of von Neumann's treatment of the natural numbers) } = The set of all natural numbers in terms of sets. Originally Posted by jsfisher Kinda funny, too, you end up with substantially the same definition for cardinality as I, just with more steps. Kinda funny, that you don't understand that your definition of non-finite set holds, only if |N| holds. Originally Posted by jsfisher A bijection is a mapping with certain properties from one set to another. Unlike your relative approach that relies on set A and another set B, I first directly define the cardinality of not non-finite set by using the same set, as follows: Definition 1: |A| = |B| iff there is bijection from A to B (where in case of bijection, B can be replaced by A and we get bijection from A to itself) So first I am focused on what happens inside set V, among its members, and it is done in order to check the validity of |V| (will be done later). v or v∪{v} are not non-finite sets, where the cardinality of v (notated as |v|) or the cardinality of v∪{v} (notated as |v∪{v}|) is defined (by definition 1) by bijection form a given domain to itself (by direct definition of cardinality). So first |v| or |v∪{v}| (which are the not non-finite members of von Neumann's treatment of the natural numbers) are defined directly. Definition 2: |A| < |B| iff (there is injection no surjection from A to B) OR (A=∅ ∧ B~=∅). ∀v ∈ V(v∪{v} ∈ V ∧ |v| < |v∪{v}|) by definition 2, which means that the expression |A| <= |B| (injection from A to B) is not satisfied from v to v∪{v} in V. Now let's look if |A| <= |B| (injection from A to B) can be used in order to establish |V|. ∀v ∈ V(v∪{v} ∈ V ∧ |v∪{v}| < |V|) In simple words: For all v∪{v} in V, |v∪{v}| is too small in order to be valued as |V|, which means that |V| can't be defined as the cardinality of all V members. In other words: |A| <= |B| (and definitely |V| <= |V|) do not hold water. As for the set of all natural numbers, |N| is not established. Here it is: Code: ```V = { ----------------------------------> Bijection according to Definition 1 | (order is irrelevant). | 0 = ∅, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 1 = { ∅ }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 2 = { ∅, {∅} }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 3 = { ∅, {∅} , {∅, {∅}} }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | v No-Bijection according to Definition 2 (order is irrelevant). ... } Order is not irrelevant since all we is that ∀v ∈ V(v∪{v} ∈ V)``` which means that < (no-bijection (by definition 2)) and = (bijection (by definition 1)) are not gathered into <= in order to define |V| <= |X|, |V| <= |V|, right inside set V, as clearly seen in the diagram above. Without |V|, there is no basis to the extension of Cardinality beyond the natural numbers. __________________ 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; Today at 02:42 AM.
 Today, 03:10 AM #533 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 Some correction of the previous post: The sentence at the end of the diagram has to be replaced by: Order is irrelevant since all we care is that ∀v ∈ V(v∪{v} ∈ V) no matter where they are in V. __________________ 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; Today at 03:12 AM.
 Today, 07:00 AM #535 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 13,243 jsfisher, you jumped all over my previous post. My previous post has to be read in the following order, in order to be understood: 1) First I use your definitions, which define the complement terms "non-finite" and "finite" (which I write as "not non-finite") 2) Then I use von Neumann's treatment of the natural numbers, which is right in line with your point, in order to define V not non-finite members, which are v and v∪{v}. 3) Then I directly define the cardinality for all v in V, by bijection (=) from v to itself. 4) Than I indirectly define the cardinality for all v and v∪{v, by injection non-surjection (<) from v to v∪{v or from ∅ to ~∅, in V. 5) (3) and (4) are done within V, at this stage. 6) Only than I use (4) in order to show that |V| is too big in order to be valued as the cardinality of set V, simply because ∀v ∈ V(v∪{v} ∈ V ∧ |v| < |v∪{v}| < |V|). Originally Posted by jsfisher Since your latest definition for "finite set" depends on your definition(s) for cardinality, you'll need to save it until later. The term is undefined until you've adequately defined cardinality. Wrong, jsfisher, in case that you have missed it, I start by using your definition of "finite set". Originally Posted by jsfisher Wrong on two counts (still). The term "non-finite member" is not (yet) defined, and your set-builder formulation falls outside what is allowed. Wrong again jsfisher, at this stage I still use your definition of the term "non-finite member". Originally Posted by jsfisher Definitions don't "hold". They simply define. Yes I already aware of your religious approach about definitions. Originally Posted by jsfisher That's still a "relative approach", and you have provided nothing to restrict it to non-finite sets. It is a "direct approach" since the bijection is done from A to itself, where v is not non-finite member (still by your definition of non-finite) of von Neumann's treatment of the natural numbers, which is right in line with your point, isn't it jsfisher? Originally Posted by jsfisher Congratulations. Your Definition 1 establishes that |V| = |V| and that for all v in V, |v| = |v| Wrong, definition 1 directly defines the cardinality of a not non-finite member v (in set V) by bijection from v to itself, where |V| is not considered at all at this stage. Originally Posted by jsfisher I doubt this is what you really mean. By this definition |V| < |V|. You need more words for the left side of the OR (and you don't need the rest of the OR at all). Again, at this stage we argue only about what is inside set V, such that v is not non-finite member (still by your definition of non-finite) of von Neumann's treatment of the natural numbers, which is right in line with your point. Originally Posted by jsfisher Why not? If |v| < |v ∪ {v}| (your definition) then there must be an injection from v to V. Wrong again, bijection (=) is from v to itself in V, where injection non-surjection (<) is from |v| to |v ∪ {v}| or form ∅ to ~∅ in V (anything about V is not considered yet, but only about what within it.) ================================================== ======= Now let's look if |A| <= |B| (injection from A to B) can be used in order to establish |V|. ∀v ∈ V(v∪{v} ∈ V ∧ |v| < |v∪{v}| < |V|) In simple words: For all v∪{v} in V, |v∪{v}| is too small in order to be valued as |V|, which means that |V| can't be defined as the cardinality of all V members. In other words: |A| <= |B| (and definitely |V| <= |V|) do not hold water. As for the set of all natural numbers, |N| is not established. Here it is: Code: ```V = { ----------------------------------> Bijection according to Definition 1 | (order is irrelevant). | 0 = ∅, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 1 = { ∅ }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 2 = { ∅, {∅} }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 3 = { ∅, {∅} , {∅, {∅}} }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, | ↓ No-bijection according to Definition 2 (order is irrelevant). | v No-Bijection according to Definition 2 (order is irrelevant). ... } Order is not irrelevant since all we is that ∀v ∈ V(v∪{v} ∈ V)``` which means that < (no-bijection (by definition 2)) and = (bijection (by definition 1)) are not gathered into <= in order to define |V| <= |X|, |V| <= |V|, right inside set V, as clearly seen in the diagram above. Without |V|, there is no basis to the extension of Cardinality beyond the natural numbers. ================================================== ======= The trick of |V| as a weak limit cardinal, is no more no less than an ad hoc artificial trick out of nowhere, which its aim it to save ZF(C) from its big crash, as already given in http://www.internationalskeptics.com...&postcount=523. __________________ 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; Today at 08:00 AM.
 Today, 04:42 PM #537 Little 10 Toes Master Poster     Join Date: Nov 2006 Posts: 2,230 Originally Posted by doronshadmi snip 1) First I use your definitions, which define the complement terms "non-finite" and "finite" (which I write as "not non-finite") There's the right way, the wrong way, and the let's-make-things-more-complicated-by-adding-junk-and-making-up-things way. Guess which way you use doronshadmi? You've already been using finite/non-finite. You only started using this nonsense term in the last 4 days. Specifically post 503. In fact, jsfisher was trying (that's a recurring theme) to correct you in using the right terms before you "write 'not non-finite". Finite and non-finite. They are complimentary. Not finite set and non-finite set, but just finite and non-finite. Let's stick with the correct terms. Stop making up words. __________________ 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.

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