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1st August 2020, 10:57 AM  #161 
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4th August 2020, 03:16 AM  #162 
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In that case, please take what I wrote immediately after definition 1 (seen in http://www.internationalskeptics.com...&postcount=160) an put it before it.

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4th August 2020, 03:37 AM  #163 
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There are three other definitions. Please arrange them in the order that you wish. No forward references please.

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4th August 2020, 04:10 AM  #164 
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5th August 2020, 06:58 AM  #165 
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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 August 2020, 07:46 AM  #166 
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"von Neumann domain" is the 'horizontal' and 'vertical' directions of the following inductive set:
Code:
{ >  ∅,  {∅},  { ∅, {∅} },  { ∅, {∅} , {∅, {∅}} },  { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, ... V } D is a placeholder for any given set. Definition 1: Cardinality is the 'size' of D iff D is at von Neumann domain. Code:
p="Cardinality is the 'size' of D" q="D is at von Neumann domain" p iff q  F F T F T F T F F T T T Code:
{ > 0 = ∅ 1 = {∅} 2 = { ∅, {∅} } 3 = { ∅, {∅} , {∅, {∅}} } 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... } Definition 4: D is called nonfinite iff D is not any particular N member (the fact that von Neumann set does not have the greatest successor, is taken, yet one is at N domain, which is equivalent to von Neumann domain). 
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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 August 2020, 09:38 AM  #167 
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Sets don't have directions. What do you mean by "von Neumann domain"? Vague pictures won't help you define it.
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9th August 2020, 05:46 AM  #168 
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I do not mean direction in terms of space, but in terms of guidance of notions.
von Neumann inductive set is not just a collection of finite sets (what I call "the 'horizontal' direction" in my diagram) but also the fact that the existence of the biggest set of this collection is not satisfied (what I call "the 'vertical downward' direction" in my diagram). Both directions are called "von Neumann domain", where being "at" means that one takes both directions of von Neumann set and not only the fact that each member is a finite set. Please look at the rest of my definitions in http://www.internationalskeptics.com...&postcount=166 . 
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9th August 2020, 11:52 AM  #169 
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Doesn't matter what type of direction you meant, sets do not have direction.
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10th August 2020, 05:51 AM  #170 
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Ok.
D is a placeholder for any given set. V is the von Neumann set of ordinals. Definition 1: Cardinality is the 'size' of D iff D is determined by V. Definition 2: N is the set of finite cardinalities iff each given n ∈ N is determined by its corresponding V member. For example: Code:
0 = ∅ 1 = {∅} 2 = { ∅, {∅} } 3 = { ∅, {∅} , {∅, {∅}} } 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... Definition 4: D is called nonfinite iff D is not any particular N member since V does not have the greatest successor. 
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10th August 2020, 06:29 AM  #171 
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10th August 2020, 07:42 AM  #172 
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Well, you can't admit (as seen in http://www.internationalskeptics.com...&postcount=139) that ZF did not establish ω or N exactly because of the bogus "all" that was arbitrarily added to it.
"all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied. Moreover, N does not exist exactly because each member of N is determined by (easily understood term) its corresponding V member (the existence of the greatest successor is not satisfied). The highlighted part is bogus since ZF did not define N as a nonfinite set (in the actual sense). The minimal set established by the Axiom of Infinity is V, such that any given N member is the cardinality of its corresponding V member. This is exactly the reason why you can't complete the following definition: N > any given n iff ... 
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10th August 2020, 08:54 AM  #173 
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doronshadmi, please fix your third message link. It does not go to the message with the quote.

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10th August 2020, 10:31 AM  #174 
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Your continual attempts to disprove definitions are pointless. Axiomatic set theories allow for infinite sets, and those sets contain "all" of there members.
Quote:

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11th August 2020, 04:19 AM  #175 
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Look jsfisher, by this categorical determination, you put the notion of definition as some kind of religious dogma.
It is not hard to understand that it is pointless to discuss with a given person about his\her dogmas. You did not show all along our discussion that, for example, ZF consistently and rigorously establishes an infinite set that contains all of its members. In order to be more concrete, please show exactly how ZF establishes ω as a limit ordinal of all smaller ordinals (https://en.wikipedia.org/wiki/Ordina...limit_ordinals):
Quote:
So, please demonstrate ZF reasoning, which actually consistently and rigorously establishes an infinite set that contains all of its members (for example: by ZF reasoning {0,1,2,3,4,5,...} is an infinite set that contains all of its members in spite of the fact that there is no largest natural number.
Quote:
Since ω is not established, the very notion of the difference between cardinal numbers and ordinal numbers, in case of infinite sets, has no basis whatsoever. 
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11th August 2020, 05:54 AM  #176 
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Interesting bit of projection, there.
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By the way, all of this has been a nice dodge by you. You still have a task of defining your version of cardinality, a task that has so far proven impossible for you. 
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11th August 2020, 07:33 AM  #177 
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Then please demonstrate exactly how ZF consistently and rigorously establishes an infinite set that includes all of its members, in spite of my lack of imagination. (b.t.w since when imagination is a factor in your arguments about actually doing mathematics?)
Well, unlike you, I do not claim that definitions can't be disproved in case that new notions about X are fundamentally changed. Actually Cantor's approach about infinite sets was an attempt to fundamentally change our understanding about them. ZF is an example of how to do math according to this attempt. But when asked to consistently and rigorously use ZF in order to establish an infinite set that includes all of its members, you simply can't do that. Instead, my lack of imagination is involved etc. ("all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied, exactly as the largest natural number does not exit, and no special kind of imagination is needed in order to understand it, just simple common sense (where being simple is far from being trivial)). You claim that ZF consistently and rigorously establishes an infinite set that includes all of its members. Your claim, your water; you carry it. 
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11th August 2020, 08:22 AM  #178 
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12th August 2020, 12:05 AM  #179 
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D is a placeholder for any given set.
∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) ) Code:
V={ ∅, {∅}, { ∅, {∅} }, { ∅, {∅} , {∅, {∅}} }, { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, ... } and it does not have the largest member. Definition 2: N is the set of finite cardinalities iff any given n ∈ N is defined by its corresponding V member. For example: Code:
0 = ∅ 1 = {∅} 2 = { ∅, {∅} } 3 = { ∅, {∅} , {∅, {∅}} } 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... Definition 4: D is called nonfinite iff D is not any particular N member since V does not have the largest member.  Well, please show consistently and rigorously, how ω is established as a limit ordinal in spite of the equivalent fact that V and N do not have their largest members (it means that the term "all" in both sets is not satisfied, where this simple fact actually prevents the existence of ω as a size (called limit ordinal) that claimed to exist by traditional mathematics as follows:
Quote:
By using only imagination one can establish whatever he/she likes, by ignoring fundamental facts (as shown above) of the considered subject. So the problem here, jsfisher, is not my "lack of imagination about the characteristics of infinite sets", but it is your inability to consistently and rigorously establish ω or N, exactly because your axiomatic framework (ZF in this case) artificially eliminates fundamental characteristics of infinite sets, where the lack of the term "all" is one of them. "ω is called limit ordinal iff ω > all n ∈ N" is no more than a string of symbols based only on imagination, without any understanding of fundamental characteristics of N as a nonfinite collection. 
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12th August 2020, 02:35 AM  #180 
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jsfisher, I wish you healthy and happy birthday.

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12th August 2020, 03:50 AM  #181 
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12th August 2020, 07:07 AM  #182 
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This is a member of Zermelo ordinals.
Please demonstrate how this member (or any other member of Zermelo ordinals) is also one of the members of V, which are constructed in terms of von Neumann ordinals ( for any given x ∈ V ( ( x ∪ { x } ) ∈ V )). It means that V has the most complex member of von Neumann construction of sets. But V does not have such member. V is defined by von Neumann construction of sets. V has at least two basic properties: (1) Any given member is finite. (2) It does not have the most complex member of von Neumann construction of sets. By (1) finite cardinality is defined in terms of natural numbers. By (2) the search for some particular finite cardinality is (potentially) nonfinite since the term "all" is not satisfied (our search continues endlessly among V members, and this is exactly the reason why ω as the limit ordinal of "all" smaller ordinals (which are V members) is not established. Since ω is not established, the difference between cardinal numbers and ordinal numbers (as done by Cantor's Transfinite system, which forces sets to be defined in terms of actual infinity (for example: "The set of all natural numbers exists")) is not established. 
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12th August 2020, 09:02 AM  #183 
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You presented two requirements for your set V. (1) It contains the empty set, and (2) for every member of V, V also contains that member's "successor".
Your requirements specify things the set V must have, but they do not identify anything as something the set must not have. They neither include nor exclude {{∅}} as a member of set V.
Quote:
For that matter, what bearing does having or not having a largest member have on this discussion? If there were a set which was both infinite and did have a "largest member" (by some appropriate definition of largest member), then what? What if there were set that had no largest member but was finite, then what? 
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12th August 2020, 12:44 PM  #184 
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Posted only for doronshadmi's convenience (bolds and italics added):
∃ I ( ∅ ∈ I ∧ ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) ) 
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12th August 2020, 12:58 PM  #185 
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13th August 2020, 06:00 AM  #186 
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For every member of V, V contains that member's "successor" that is defined by the union of x with {x} (written as x ∪ {x}).
Please define {{{}}} as the union of x with {x}. As seen in case of V members' construction, they become more and more complex (where the least complex member is {}), yet V does not have the most complex member, by this members' construction. Please define such set, but please remember that its members are pure sets (constructed only by "{"..."}" pairs, at the base level of sets' representations). Again, please think about nonempty sets that its distinct members are pure sets (constructed only by "{"..."}" pairs, at the base level of sets' representations). By ∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) ) we get V members. By ∃ X ( ∅ ∈ X ∧ for any given x ∈ X ( { x } ∈ X ) ) we don't get V members (except ∅ that is not a successor in both constructions, and it is specifically included in both sets). You see jsfisher, you take V as a placeholder for any set, but V has a unique construction, which is different than X unique construction. Please look at http://www.internationalskeptics.com...&postcount=179 . As you can see there, D is a placeholder for any set, but V is a unique set that is used to define cardinality, whether it is finite or (potentially) nonfinite (since its largest member does not exist). 
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13th August 2020, 08:44 AM  #187 
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The requirement you have cited does not rule out {{∅}} from being in V. It does guarantee that if {{∅}} be a member, then so must { {{∅}}, {{{∅}}} }, but it is silent as to whether {{∅}} is a member in the first place.
Quote:

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13th August 2020, 05:17 PM  #188 
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13th August 2020, 06:57 PM  #189 
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Yeah, that is all true. The default font size this forum uses doesn't help much either.
Here it is at SIZE = 5: ∃I (∅∈I ∧ ∀x∈I ((x∪{x})∈I)) I also considered substituting brackets for parenthesis at one place to alternate the grouping symbols. Something like this: ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) The bold isn't all that helpful at SIZE = 5, but braces are more obvious. Switching to Times New Roman: ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) YMMV 
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13th August 2020, 08:13 PM  #190 
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13th August 2020, 08:16 PM  #191 
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14th August 2020, 04:02 AM  #192 
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∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) ) has the following members according to its given construction:
Code:
V={ ∅, {∅}, { ∅, {∅} }, { ∅, {∅} , {∅, {∅}} }, { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, ... } Code:
X={ ∅, {∅}, {{∅}}, {{{∅}}}, {{{{∅}}}}, ... } No steps (from the Platonist point of view) the unique construction of V or X sets is taken at once (or in parallel, if you wish). From the, so called, constructionist point of view, it may be taken serially. Some common properties of both V and X sets are: 1) No successor of x is a member of x (in order to be considered as a successor, in the first place) 2) Any given member of X or Y is a finite set. 
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14th August 2020, 06:06 AM  #193 
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Some corrections of my previous post:
2) Any given member of V or X is a finite set. 3) No successor is the largest member of V or X. 
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14th August 2020, 06:12 AM  #194 
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14th August 2020, 06:13 AM  #195 
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"Ideas are also weapons."  Subcomandante Marcos "We must devastate the avenues where the wealthy live."  Lucy Parsons "Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!"  Mikhail Bakunin 

14th August 2020, 09:10 AM  #196 
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That would be the Axiom on Infinity. It stipulates the existence of a set (that's the ∃V part) with two properties. It does not, however, tell us specifically what that set is.
Quote:
Doronshadmi, I have a bag of marbles. All of my blue marbles are in the bag. Is there a red marble in the bag? Answer: From the information I've given you, you cannot tell. Nor can you tell if {{∅}} be a member of the set stipulated by the Axiom of Infinity. 
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15th August 2020, 09:05 PM  #197 
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16th August 2020, 06:22 AM  #198 
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∃ N ( ∅ ∈ N ∧ for any given x ∈ N ( ( x ∪ { x } ) ∈ N ) )
Since "base elements [predecessors], and their successors" are defined by the axiom of infinity, N members are ordered, otherwise the term successor is not used, in the first place. Moreover, x is the immediate predecessor of ( x ∪ { x } ) and ( x ∪ { x } ) is the immediate successor of x, so the way of how any given nonempty x member is defined, depends on how the successor of any given nonempty x is defined (they are its distinct foot prints that it leaves 'behind'). Definition 1: Set Y is called the immediate successor of set X iff X is the set of all the predecessors of Y AND Y is not a member of X (where the term all is valid only if X is a finite set, as will be shown in the case of set N). So set N has ∅ as its base member, which is the predecessor of any given nonempty set that is a member of set N. Since order is important (otherwise the terms predecessor or successor are not used, in the first place) N members (by the axiom of infinity) are as follows: Code:
N={ ∅ (the base member), ∅∪{∅}={∅} , {∅}∪{{∅}}={ ∅, {∅} }, {∅,{∅}}∪{{∅,{∅}}}={ ∅, {∅} , {∅, {∅}} }, {∅,{∅},{∅,{∅}}}∪{{∅,{∅},{∅,{∅}}}}={ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, ... }  (B.T.W the largest member of a given set, is the member of that set, which has no successor. 
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16th August 2020, 06:44 AM  #199 
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doronshadmi, if you are going to use the Axiom of Infinity, quote it correctly. This shows your level of credibility. It has been provided to you several times on this page alone.
∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) 
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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. 

16th August 2020, 10:09 AM  #200 
ETcorngods survivor
Moderator Join Date: Dec 2005
Posts: 22,947


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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 

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