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27th July 2020, 06:19 AM  #121 
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Now here is my question to you:
Please complete the following definition: N > any given n iff ... 
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27th July 2020, 06:53 AM  #122 
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27th July 2020, 06:56 AM  #123 
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28th July 2020, 12:06 AM  #124 
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D is a placeholder for sets, therefore D is not any particular set, so I do not have to provide means to determine what the bus will do for a given D.
If the bus stands (D is determined by some n) then D is finite. If the bus does not stand (D is not determined by some n) then D is nonfinite. 
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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. 

28th July 2020, 12:17 AM  #125 
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Because by traditional mathematics N > any given n
Since you represent here the reasoning of traditional mathematicians about the discussed subject (where by traditional mathematics N > any given n), you are able to complete the following definition: N > any given n iff ... 
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28th July 2020, 04:10 AM  #126 
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28th July 2020, 04:13 AM  #127 
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28th July 2020, 05:09 AM  #128 
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Let D be a placeholder for any set.
Let N be the set of natural numbers. Definition 1: Cardinality is the "size" of D (notated as D) iff it is related to the members of N, where 0 is a member of N. "related to" means that Cardinality is not necessarily some particular N member, yet it is at the domain of N, where N does not have the greater successor. 
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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. 

28th July 2020, 06:28 AM  #129 
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I have a set D which is the set of combinations of the English vowels (a, e, i, o, u). How do I determine the set's cardinality?

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28th July 2020, 06:58 AM  #130 
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By the number of the combinations that can be done by 5 distinct things (the English vowels, in this case).
Please see https://en.wikipedia.org/wiki/Combination 
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28th July 2020, 07:07 AM  #131 
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The element aa is an acceptable combination.
How do you determine the set's cardinality? 
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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. Last edited by Little 10 Toes; 28th July 2020 at 09:00 AM. Reason: Bolded the element aa; corrected the question 

28th July 2020, 09:11 AM  #132 
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If aa is an element (https://en.wikipedia.org/wiki/Element_(mathematics)) of D, than D is at least 1.
Generally, you do not provide enough information about your set (which its members are combinations) since you did not determine k (see https://en.wikipedia.org/wiki/Combination). 
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28th July 2020, 09:22 AM  #133 
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That didn't cut it as a definition that last time you posted it. Reposting it now doesn't improve on that.
The definition body itself is "grammatical" nonsense. You have presented "Cardinality is the 'size' of D" as a true/false proposition. So, in some cases cardinality is the "size" of D, and in other cases it is not. Great, I suppose, but what does cardinality actually mean? As for "'related to' means that Cardinality is not necessarily some particular N member", you haven't described with any specificity what 'related to' means. From your definitions, and only your definitions, with no extra handwaving or undefined filler, how would one determine the cardinality of some set, S? From your definitions, and only your definitions, with no extra handwaving or undefined filler, how would one determine the cardinality of a set, {a,b,c,d}, to be 4 and not be 21? 
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28th July 2020, 09:40 AM  #134 
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28th July 2020, 09:04 PM  #135 
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Please note doronshadmi, I asked the same question three times. Yes, there are ways to determine combinations. Your answer of, "then D is at least 1" is nonsense. Why? Because that was not what I asked for. I will ask it again very clearly.
In my example, how do you determine the cardinality of set D? 
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28th July 2020, 11:48 PM  #136 
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Let N be the set of natural numbers.
Cardinality is the 'size' of D, whether it is determined by a particular N member (in this case D=particular n and D is called finite set) or it is determined according the fact that N does not have the greatest successor (in this case D is not any particular n, yet we stay at N domain and D is called nonfinite set). So being at N domain this is the true state of D, whether it is determined by some particular n, or not. Code:
p="Cardinality is the 'size' of D" q="D is at N domain" p iff q  F F T F T F T F F T T T Definition 1: Cardinality is the 'size' of D (notated as D) iff it is related to the members of N, where 0 is a member of N. "related to the members of N" simply means that D is at N domain, whether it is determined by some particular n, or not (in this case the fact that N does not have the greatest successor is taken). The definitions in http://www.internationalskeptics.com...7&postcount=98 have nothing to do with defining specific values of cardinality, so your question simply show that you basically does not follow after what they are actually define. 
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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. 

29th July 2020, 04:14 AM  #137 
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29th July 2020, 06:07 AM  #138 
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No jsfisher, my definition of cardinality is sufficient to determine the concept of cardinality, which is not limited to your restriction to define some particular cardinality by using the definition.
It is called generalization, which enable to use it in order to distinguish between finite and nonfinite sets, without the need for special axioms (which in case of ZF axiom of infinity and the axiom schema of restricted comprehension, they do not provide a nonfinite set that its cardinality > any given N member, but such a set is arbitrarily determined. Also the very notion of axiom schema as a nonfinite set of axioms, is arbitrarily determined). 
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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. 

29th July 2020, 09:15 AM  #139 
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The best that the ZF axiom of infinity and the axiom schema of restricted comprehension (with more ZF axioms) establish, is the inductive set known by the name "von Neumann ordinals" that does not have the greatest successor, so the term all in the following equate is arbitrarily plugged in:
Now let's look at this: This is a concrete example of an arbitrary expression.  One may ask if N > the cardinality of any given N member (where in our case, the members of N are von Neumann ordinals, as constructed by based on the two properties of ZF axiom of infinity). Since the cardinality of any given member of N is finite and the term all is arbitrarily plugged in (bye bye to the fact that N does not have the greatest successor, which actually prevents the term all) N > cardinality of any given N member, is inevitable by ZF. Please look at your reply in http://www.internationalskeptics.com...&postcount=123 and the following quote: The cardinality of any given N member (where N is actually the set of von Neumann ordinals) can't be but some natural number, so "not a requirement of mine" has no basis whatsoever. Therefore please complete the following definition (by using ZF reasoning): N > any given n iff ... 
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29th July 2020, 09:47 AM  #140 
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29th July 2020, 09:59 AM  #141 
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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. 

29th July 2020, 10:03 AM  #142 
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No, it is based on iff (your requirement) as seen in http://www.internationalskeptics.com...&postcount=136.

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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. 

29th July 2020, 02:04 PM  #143 
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That's the best? You seem to have overlooked just how foundational set theory is to the entire body of Mathematics.
Quote:
Stop with the countless diversions, please. You've made claims. You need to defend your claims, and not jump to any topic you can that isn't among your claims. Focus: Start with your definition of cardinality as you are using the term. 
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29th July 2020, 02:21 PM  #144 
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I have no such requirement.
If there is something to be defined that is boolean valued, say, for example, whether a set is nonfinite, then an ifandonlyif construct is perfectly reasonable. A set is nonfinite if and only if something else (that is already welldefined) is true. But you aren't expressing things as boolean valued. An ifandonlyif construction makes absolutely no sense for that, except perhaps to someone who really doesn't understand basic mathematical logic. Moreover, you want the cardinality of some sets to be whole numbers, but the gibberish you presented doesn't even begin to define cardinality in a way that the cardinality of a set can be determined. 
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29th July 2020, 02:23 PM  #145 
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30th July 2020, 02:43 AM  #146 
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No, it is at the heart of our discussion.
By trying to force "all" on a set that the existence of its greatest successor is not satisfied, you actually eliminate the very existence of an endless collection (where a set is some particular case of collection). For example, please define the greatest successor of the set of von Neumann ordinals. 
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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. 

30th July 2020, 04:05 AM  #147 
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Very simple:
Definition 1: Cardinality is the 'size' of D (notated as D) iff it is related to the members of N, where 0 is a member of N. "related to the members of N" simply means that D is at N domain, whether it is determined by some particular n, or not (in this case the fact that N does not have the greatest successor is taken). D={a,b,c,d} D={a,b,c,d} (the 'size' of D) By constricting the natural numbers in terms of sets, as done by von Neumann {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}} = 4 Since {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}} and {a,b,c,d} are bijective, D can't be but 4. 
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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. 

30th July 2020, 04:12 AM  #148 
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30th July 2020, 04:43 AM  #149 
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There is no leap.
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)) ). The highlighted part is bijection. As for von Neumann construction, natural numbers and Cardinality: Code:
{ > 0  = ∅ 1  = {∅} 2  = { ∅, {∅} } 3  = { ∅, {∅} , {∅, {∅}} } 4  = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... V } The vertical downward direction is nonfinite cardinality (the existence of the greatest successor is not satisfied). In both cases you do not leave 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. 

30th July 2020, 04:55 AM  #150 
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30th July 2020, 06:07 AM  #151 
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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. 

30th July 2020, 06:11 AM  #152 
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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. 

30th July 2020, 06:48 AM  #153 
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30th July 2020, 08:10 AM  #154 
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Let's try that out:
Let A = {a, b, c}, and let B = {}. A = B ? Well, impossible to tell from Doronshadmi's "definitions", but I will assume Doronshadmi would say "false" since he wants A to be 3 and B to be 0. (We'd reach the same final result were the answer "true".) B = A ? "False". Therefore: A = B iff B = A is "true". Conclusion: Each element of A is mapped to exactly to one element of B and each element of B is mapped to exactly to one element of A. 
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31st July 2020, 12:29 AM  #155 
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Thank you, you are right.
"Each element of A is mapped to exactly to one element of B" is unclear since, for example, it can be also surjection, and so is the case about the other direction. So, back to work: 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 As about von Neumann domain, natural numbers and Cardinality: Code:
{ > 0  = ∅ 1  = {∅} 2  = { ∅, {∅} } 3  = { ∅, {∅} , {∅, {∅}} } 4  = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... V } The vertical downward direction is nonfinite cardinality (the existence of the greatest successor is not satisfied). In both cases one does not leave von Neumann domain.)  Definition 2: N is the set of finite cardinalities iff any given n ∈ N member is determined by a particular von Neumann member ( as follows: ) 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 we do not leave N 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. 

31st July 2020, 03:52 AM  #156 
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Definition 2 correction:
Definition 2: N is the set of finite cardinalities iff each given n ∈ N is determined by each von Neumann member ( as follows: ) Code:
{ > 0 = ∅ 1 = {∅} 2 = { ∅, {∅} } 3 = { ∅, {∅} , {∅, {∅}} } 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... } 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

31st July 2020, 06:55 AM  #157 
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Definition 2 correction:
Definition 2: N is the set of finite cardinalities iff each given n ∈ N is determined by its corresponding von Neumann member ( as follows: ) Code:
{ > 0 = ∅ 1 = {∅} 2 = { ∅, {∅} } 3 = { ∅, {∅} , {∅, {∅}} } 4 = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... } 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

31st July 2020, 08:44 AM  #158 
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Doronshadmi, please start using different set letters. N is usually used for the set of natural numbers.

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31st July 2020, 04:38 PM  #159 
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Yes.
Quote:
The whole point of my post was that you are misusing the "if and only if" construct. Whatever it is you think what you write might mean, you are making it gibberish by what you do with "iff". There is another thing you need to do, too: Your "definitions" must not have forward references. Your definitions may not rely on any terms that aren't yet defined. If Definition 1 depends on Definition 4, for example, then you need to reorder your definitions to move Definition 1 to a new spot after Definition 4. 
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31st July 2020, 10:08 PM  #160 
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Ok here is my corrected version.
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 As about von Neumann domain, natural numbers and Cardinality: Code:
{ > 0  = ∅ 1  = {∅} 2  = { ∅, {∅} } 3  = { ∅, {∅} , {∅, {∅}} } 4  = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} } ... V } The vertical downward direction is nonfinite cardinality (the existence of the greatest successor is not satisfied). In both cases one does not leave von Neumann domain.)  Definition 2: N is the set of finite cardinalities iff each given n ∈ N is determined by its corresponding von Neumann member ( as follows: ) 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 we do not leave N 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. 

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