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2nd July 2020, 06:38 AM  #41 
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jsfisher, is it possible to define Cardinality only by <= relationship from set A to set B (written as A <= B)?
If no, then cardinality is a part of the definition of Cardinality as shown, for example, in http://www.internationalskeptics.com...4&postcount=40 . If you disagree with me, then please air your view about this post. Thank you. 
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2nd July 2020, 06:44 AM  #42 
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2nd July 2020, 06:56 PM  #43 
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As zooterkin properly points out, I did not define cardinality. I defined a relationship.
I chose "<=" as the symbol for this dyadic boolean relationship. Cardinality of a set is normally indicated by vertical bars to either side of the set name  a convention so well known to almost everyone that explicit mention of the convention should not be required. In context, A and B would be interpreted as symbols for arbitrary sets  another convention for which no explicit mention should be necessary. To define the relationship, it must be presented with operands so the operands can participate in the definition. Ergo, "A <= B" is the thing being defined. I.e., "the relationship symbolized by <= with operands, A and B". And "there exists an injection from A to B" is the definition. So, once again, cardinality is not being defined, but a relationship between the cardinality of two sets is being defined. This is not third grade math, so please stop trying to apply third grade concepts to it. 
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3rd July 2020, 06:06 AM  #44 
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Thank you to zooterkin and jsfisher. I knew what I was trying to say, I just wasn't the right terms.

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4th July 2020, 07:18 AM  #45 
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Please look at http://www.internationalskeptics.com...postcount=3448 do you see there that jsfisher literally stats that he defines ‘relative cardinality’?

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4th July 2020, 07:26 AM  #46 
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4th July 2020, 08:26 AM  #47 
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Sorry jsfisher, here is what you have said:
No mention of a definition of Relative cardinality, isn't it? So, after some number of posts and a little help from zooterkin, your definition of Cardinality, is not a definition of Cardinality, but it is actually a definition of a relative measure between the cardinalities of two arbitrary sets. In other words, your determination of the need to be clear is not fulfilled.  Any way, 'The definition of a relative measure between the cardinalities of two arbitrary sets' has a motivation behind it, which is: To be general enough, in order to not distinguish between finite or infinite sets. ZF(C) is actually a theory of sets, where its Axiom Of Infinity, is not actually Axiom Of Infinity unless more things are involved with it. This style of work is done for the name of generalization, such that any given statement does not stand for its own, but depends on the interpretation that is given to it according to the rules of a given context. In other words, traditional mathematics is based on the of Philosophy of relativism (https://plato.stanford.edu/entries/relativism/) and jsfisher actually demonstrates it, no matter what he argues about the independence of traditional mathematics in any philosophical issues. 
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4th July 2020, 10:13 AM  #48 
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Yes jsfisher, this section is about Religion and Philosophy, in case that you have missed it.
Furthermore, what is called traditional mathematics is deeply rooted in Philosophy, as shown in http://www.internationalskeptics.com...2&postcount=47 . 
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4th July 2020, 10:29 AM  #49 
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Now back to the current issue, my answer to http://www.internationalskeptics.com...postcount=3487 is given in http://www.internationalskeptics.com...8&postcount=31 .
According to a reasonable criticism (that unlike you, is aware of the philosophical point of view of it) it will be changed. If you still reject Philosophy as a source of motivation for mathematical work, any mathematical work, then there is no real discussion between us, because all you care is to show how Philosophy is disconnected from actual mathematical work (which is by itself, no more no less than some philosophical point of view). Your particular philosophical view of mathematics is actually like a string that penetrates trough all your posts since our first dissuasion that was done almost 12 years ago. I have learned a lot from your particular philosophical view of mathematics, and tried to air my particular philosophical view of mathematics, which unlike your philosophical point of view, does not reject philosophy as an actual factor of mathematical work, any mathematical work. 
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4th July 2020, 03:24 PM  #50 
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You are just now figuring out that the whole point of building Mathematics on a axiomatic basis is to be disconnected, as you say, from Philosophy? You need to pay more attention. Philosophy credits itself with having "really good ideas"^{TM}, which it may or may not have, but it does not have the consistency nor any sort of rigor essential for advancing Mathematics. The Dark Ages are over; time you moved on. 
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5th July 2020, 01:15 AM  #51 
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You can move on as much as you like, Philosophy is an essential factor of any mathematical, as very simply demonstrated in http://www.internationalskeptics.com...2&postcount=47 .

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5th July 2020, 01:33 AM  #52 
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This is no less no more than your philosophical point of view about the relations between Mathematics and Philosophy.
This approach of separated islands of context dependent mathematical frameworks, is a direct result of the understanding that most of these separated and, so called, rigor contexts dependent islands (which are able to deal with Arithmetic) can't prove their own consistency by finitely many axioms (the end of Hilbert's program, as a result of Godel's incompleteness theorems). The end of Hilbert's program, is one of the reasons why mathematicians stopped dealing with "big ideas" (philosophical notions) and reduced their work to "small ideas" (the technical mechanism of rigorous methods).  Please critisize http://www.internationalskeptics.com...8&postcount=31 and it's links, by your, so called, rigorous mathematical working methods. 
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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 July 2020, 02:17 AM  #53 
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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 July 2020, 02:32 AM  #54 
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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 July 2020, 07:50 AM  #55 
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Please provide evidence of this claim.

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5th July 2020, 09:48 AM  #56 
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He's right about that consistency thing. Gödel’s second Incompleteness Theorem proved that any consistent formal system in Mathematics rich enough to include arithmetic cannot prove its own consistency.
Be that as it may, though, the point is irrelevant in the current discussion arc. Mathematics shuns inconsistency even though it cannot usually guarantee its absence while Philosophy cherishes it. 
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6th July 2020, 04:25 AM  #57 
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It is relevant, as shown in http://www.internationalskeptics.com...4&postcount=52 , and my argument is not about Mathematics vs. Philosophy, but it is about the philosophical motivations that actually deeply influence on the rigorous mechanism of any mathematical work.
As for the current discussion Zermelo and his followers vary carefully established a theory of sets such that it will preserve Cantor's notions of infinite sets in terms of actual infinity. For example, Cardinality is not defined directly by observations of natural numbers (as done for example in http://www.internationalskeptics.com...8&postcount=31, and was improved by jsfisher's criticism) , but it is defined indirectly through a relative measurement (a onetoone mapping in this case) from the members of one set (called A) to the members of another set (called B), in such a way that enables to plug in Cantor's transfinite system (which is actually the translation of a philosophical notion of infinity in terms of a set that actually has infinitely many members that are taken as a complete whole). By this indirect definition of Cardinality one enables to suddenly jump into a measurement unit like aleph0, like a third degree magician, which truly believes that that by indirect definition of Cardinality the audience will not pay attention to his inconsistent trick. Unlike jsfisher indirect approach, which also ignore the philosophical motivations behind his indirect definition of cardinality, http://www.internationalskeptics.com...8&postcount=31 does not need any indirect methods in order to define Cardinality, and I do not try to hide my philosophical motivations, which argue that any infinite collection (where an infinite set is a particular case of an infinite collection) is no more than potentially infinite since its accurate size is by definition not satisfied. By going beyond the notion of collections, and by using philosophical direct approach of the discussed subject, actual infinity is essentially noncomposed, as already demonstrated in http://www.internationalskeptics.com...postcount=3285 . 
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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. 

14th July 2020, 11:22 PM  #58 
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Let's move on jsfisher.
So according to traditional mathematics: 1) Cardinality is a relative measure of "size" of sets where A <= B if and only if there exists an injection from A to B. 2) The ZF(C) axiom of infinity has two properties: Property 1: ∅ ∈ I Proeprty 2: ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) Please (by using (1),(2) and (if needed) more ZF(C) axioms) rigorously establish I as an infinite set (which means that its cardinality is strictly greater than any natural number (where 0 is also taken as a natural number)). 
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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. 

15th July 2020, 07:52 AM  #59 
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Why? If you read the Axion of Infinity, it already postulated that the set I is infinite.

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15th July 2020, 04:53 PM  #60 
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Actually, it doesn't. The Axiom establishes that there exists a set meeting two properties. You need to define what it means for a set to be infinite before you can establish that that set meets the definition.
The normal approach to that is to adopt a measure of set size, and then it follows. 
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15th July 2020, 05:09 PM  #61 
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No, cardinality, as a relative measure of set size, can be defined by defining a relationship between the sizes of two sets. Id est, A <= B if and only if there exists an injection from A to B.
Quote:
Quote:
Why should I do that? You are the one trying to discredit "traditional mathematics." The burden to prove or disprove whatever it is that establishes your claim lies with you. 
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15th July 2020, 10:19 PM  #62 
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jsfisher, here is your definition of Cardinality as you wrote in http://www.internationalskeptics.com...postcount=3448.
So why do you say No?  Any way: 1) The definition of Cardinality as a relative measure of set size that can be defined by defining a relationship between the sizes of two sets: A <= B if and only if there exists an injection from A to B. Ok thank you. 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 ) Jsfisher, please move on by consistently and rigorously establish the highlighted parts, which have been taken from your quote. 
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16th July 2020, 04:47 AM  #63 
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You are the one with something to prove (or disprove), not me. These attempts of yours to reverse responsibilities or divert to side topics do not advance your cause.
It would be especially good for you to start with a statement of just what is it you are claiming  a nice, simple declaration of your treatise. Is that too much to ask? 
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17th July 2020, 05:18 AM  #64 
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So at this stage of our discussion you do not "have the consistency nor any sort of rigor essential for advancing Mathematics" about the highlighted parts taken from your quote (seen in http://www.internationalskeptics.com...2&postcount=62).
Actually, "The Dark Ages are" not "over" for you as long as you don't consistency and rigorously support your own highlighted parts (seen in http://www.internationalskeptics.com...2&postcount=62). In order to actually do that, please consistency and rigorously support your own highlighted parts (seen in http://www.internationalskeptics.com...2&postcount=62). 
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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. 

17th July 2020, 06:01 AM  #65 
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17th July 2020, 08:24 AM  #66 
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It would be especially good for you to consistency and rigorously (which is, as you say, "essential for advancing Mathematics") establish the highlighted parts, which have been taken from your quote (seen in http://www.internationalskeptics.com...2&postcount=62).
Is that too much to ask? 
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17th July 2020, 09:42 AM  #67 
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17th July 2020, 11:54 AM  #68 
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Yes, it is. I have nothing by itself to prove in this thread, Doronshadmi. You do. You started the thread, after all, presumable to make some point.
It would be especially good for you to start with a statement of just what is it you are claiming  a nice, simple declaration of your treatise. Is that too much to ask? 
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17th July 2020, 09:58 PM  #69 
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jsfisher you, of course, free to do whatever you like.
Until this very moment you did not address those quotes consistently and rigorously, which is, as you say, "essential for advancing Mathematics". My deceleration is this: Nothing is normal (where in this context normal means: consistently and rigorously) of how traditional mathematics "adopt a measure of set size" in terms of the nonfinite, so nothing follows because it is arbitrarily done by it. If you disagree with my deceleration, please consistently and rigorously support your quotes above. No one will take away from you the freedom to evade my deceleration. But your evasion will be noted. 
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18th July 2020, 08:07 AM  #70 
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That is not how that works.
You have a claim. You provide evidence that you are right. You have not provided evidence that you/your belief system is/are right. And you are not accurately quoting jsfisher. His full quote is
Quote:
If you disagree with jsfisher's claim, challenge him on it and ask for proof/evidence. If you want, you can provide evidence that philosophy DOES have consistency and rigor essential for advancing Mathematics. 
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18th July 2020, 08:10 AM  #71 
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Just to be clear doronshadmi, your issue is about how mathematics defines cardinality? Nothing else, just how cardinality is defined? A simple Yes/No answer is fine.
Edit: Once your viewpoint is clear, it can be addressed. 
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18th July 2020, 09:51 AM  #72 
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So? The outofcontext quotation isn't an attempt to advance Mathematics.
If you wish to challenge my statement, then feel free to do so, but include some basis for your objection that can become the basis for discussion (if anyone cares to engage).
Quote:
Your treatise is difficult to parse. Do not introduce nonstandard usage of a word like 'normal' if you mean something else. Use those somethingelse words. Your bad habit of continually adding unnecessary words only muddles you posts. It appears you are suggesting that there is something wrong with Mathematics for having ways to measure the size of a set. Is that right? You would prefer cardinality not be defined? 
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18th July 2020, 11:36 PM  #73  
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I claim that the measure of the size of a nonfinite set, is not done consistently and rigorously (which means that it is done arbitrarily) by traditional mathematics.
In http://www.internationalskeptics.com...1&postcount=43 you explicitly state that So, cardinality is not being defined by traditional mathematics. In that case, why do you asking me if I "would prefer cardinality not be defined?" when you are explicitly state that (by traditional mathematics) "cardinality is not being defined"? Your bad habit to avoid your own standards about traditional mathematics produces questions that actually must be answered by you.  As for me, I prefer cardinality to be defined without eliminating the difference between finite and nonfinite cardinality, for example: 1) I do not need iff in order to define Cardinality by the members of the set of natural numbers (notated as N), where natural numbers (including number 0) are naturally understood (no extra maneuvers are needed). 2) I provide symbols to the concepts "less than", "equal to", "greater than" (which I tend to replace by "more than") which are naturally understood (no extra maneuvers are needed). 3) I use iff in a vary simple way in case of relative measure, between two arbitrary sets A and B, where their cardinalities (what is compared) and the terms of how they are compared are simply and intuitively addressed, because of the simple use of iff. 4) By being simple and intuitive in definitions 1 and 2, I am able to very simply define finite (definition 3) and nonfinite (definition 4) sets, without any need to add anything, which are not already given by definitions 1 to 4. 5) Definition 5 very simply addresses nonstrict inequality as a range between strict inequality (< or >) and equality (=). Once again you are invited to criticize what is written in this post and in http://www.internationalskeptics.com...9&postcount=15 post. Thank you.


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18th July 2020, 11:47 PM  #74 
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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. 

19th July 2020, 07:09 AM  #75 
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19th July 2020, 07:21 AM  #76 
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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. 
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20th July 2020, 03:46 AM  #77 
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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. 

20th July 2020, 04:07 AM  #78 
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20th July 2020, 04:59 AM  #79 
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Your evasion has been noted once again.
Do you see this, jsfisher?
Originally Posted by jsfisher
Originally Posted by jsfisher
Originally Posted by jsfisher
 The highlighted quotes are no more than philosophical declarations, and there is here in this thread a person that says that he can address those declarations consistently and rigorously, which is "essential for advancing Mathematics". Until this very moment this person did not consistently and rigorously established a nonfinite set, and in order to avoid it, he tries to represent the discussion in terms of some kind of personal discussion, as clearly seen in the first quote of this post. His evasions are noted time after time. 
Originally Posted by jsfisher

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20th July 2020, 05:05 AM  #80 
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Twelve years and still going...

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