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20th July 2020, 09:23 AM  #81 
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20th July 2020, 11:56 PM  #82 
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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 nonfinite 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 nonfinite 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 nonfinite as long as a nonfinite 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 nonfinite set in a way that is not arbitrary, exactly because cardinality is not directly defined by it. I do not have to guess, as long as your system does not define Cardinality directly, you can't consistently and rigorously establish a nonfinite set. 
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21st July 2020, 04:11 AM  #83 
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Yes, very true. My approach to cardinality applies to all sets, not just some.
Quote:
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 'nonfinite set'. And how could that be remedied? A simple definition, perhaps, of what it means for a set to be nonfinite? Maybe something like: A set, X, is nonfinite if and only if N <= X, where N is the von Neumann ordinal.Other definitions are possible, but this one will do.
Quote:

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21st July 2020, 04:43 AM  #84 
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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 nonfinite set (after all we discuss about mathematics, isn't it?) Please provide another definition or consistently and rigorously show that N is a nonfinite set. Please define reality. 
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21st July 2020, 04:49 AM  #85 
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21st July 2020, 04:59 AM  #86 
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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. 

21st July 2020, 05:05 AM  #87 
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21st July 2020, 05:29 AM  #88 
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A Wikipedia quote (https://en.wikipedia.org/wiki/Ordina...on_of_ordinals):
Quote:
Quote:

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21st July 2020, 05:31 AM  #89 
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Sorry but your link does not work.
If you mean to "A set, X, is nonfinite 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. 
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21st July 2020, 08:15 AM  #90 
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Please define "nonfinite set". Because you do not use the term "infinite set", your term must have a special meaning to you.

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21st July 2020, 08:43 AM  #91 
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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 "nonfinite set" is sufficient to show the von Neumann ordinal as a nonfinite 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. 
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21st July 2020, 08:58 AM  #92 
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I apologize. It seems I have introduced some confusion I did not intend by using the term "von Neumann ordinal" in a nonstandard 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 nonfinite set if and only if N <= X, where N is the minimal set established by the Axiom of Infinity. 
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22nd July 2020, 02:35 AM  #93 
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ZF Axiom of Infinity is a good example of how Traditional Mathematics arbitrarily forces N to actually be nonfinite 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 nonfinite set if and only if N <= X" is simply nonsense. Here is how this nonsense is done: ...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 nonfinite 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. 
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22nd July 2020, 02:56 AM  #94 
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Please read very carefully http://www.internationalskeptics.com...9&postcount=73 including its link.

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22nd July 2020, 04:48 AM  #95 
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22nd July 2020, 04:53 AM  #96 
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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 selfserving philosophy, I suppose, but not particularly useful nor convincing. 
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22nd July 2020, 05:50 AM  #97 
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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. 

22nd July 2020, 06:44 AM  #98 
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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 nonfinite sets. So: Definition 3: Set A is called finite iff given any n in N, A is any particular n Definition 4: Set A is called nonfinite 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 nonfinite 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 nonstrict inequality iff ((A ≤ B) iff (B ≥ A)). 
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22nd July 2020, 10:10 AM  #99 
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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 ifandonlyif is taken as a constraint requiring the lefthand side (the thing being defined) bear the same truth value as the righthand side.
You are claiming that the lefthad side, "Cardinality is the 'size' of A", is a statement that can be either true or false. On the other hand, the righthand 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 "nonfinite set"? 
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23rd July 2020, 12:31 AM  #100 
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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). 
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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. 

23rd July 2020, 02:39 AM  #101 
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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. 
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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. 

23rd July 2020, 04:13 AM  #102 
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23rd July 2020, 04:50 AM  #103 
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http://www.internationalskeptics.com...&postcount=100
Please look at definitions 3 and 4 in http://www.internationalskeptics.com...7&postcount=98. 
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23rd July 2020, 06:19 AM  #104 
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23rd July 2020, 07:16 AM  #105 
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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. 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 nonfinite is not actually be infinite, as give by the following definition: Definition 4: Set A is called nonfinite iff given any n in N, A is not any particular n Please look at definitions 3 and 4 in http://www.internationalskeptics.com...7&postcount=98. 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. 
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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. 

23rd July 2020, 07:35 AM  #106 
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By going beyond the notion of collections (where sets are particular forms of collections) actual infinity is essentially noncomposed, as already demonstrated in http://www.internationalskeptics.com...postcount=3285.

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23rd July 2020, 09:52 AM  #107 
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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 
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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. 
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24th July 2020, 02:51 AM  #109 
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Definition 3 simply states that D is finite iff D is a particular n.
Definition 4 simply states that D is nonfinite 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. 
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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. 

24th July 2020, 05:49 AM  #110 
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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. 

24th July 2020, 10:06 AM  #111 
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24th July 2020, 10:14 AM  #112 
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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.) 
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26th July 2020, 06:59 AM  #113 
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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 nonfinite 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: 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. 
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26th July 2020, 07:34 AM  #114 
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Given a set D that is finite, which particular member of N would that be?
Quote:
Quote:
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. 
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27th July 2020, 02:45 AM  #115 
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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 nonfinite set (all we care is that the bus does not stop, and in this case D is nonfinite 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 nonfinite). So zooterkin question is irrelevant, in this case. You may say that (D is finite) OR (D is nonfinite) is a tautology, but this tautology is found also by traditional mathematics. But traditional mathematics define N as > any given n without any basis. 
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27th July 2020, 04:39 AM  #116 
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27th July 2020, 04:54 AM  #117 
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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 nonfinite (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. 
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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. 

27th July 2020, 05:18 AM  #118 
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27th July 2020, 05:54 AM  #119 
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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 nonfinite. That's all we need in order to distinguish between the finite and the nonfinite. No decision of involved in order to distinguish between stands or does not stand. 
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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. 

27th July 2020, 06:03 AM  #120 
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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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