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Old 21st June 2020, 07:55 AM   #1
doronshadmi
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Deeper than primes - Continuation 1/3*9

Originally Posted by jsfisher View Post
|A| <= |B| if and only if / is defined by / means there exists an injection from A to B
You wrote
Originally Posted by jsfisher View Post
Given some set A and some set B, |A| <= |B| is a proposition that may or may not be true
The focus is only on |A| <= |B| proposition, which can be written also as (|A| < |B|) OR not(|A| < |B|) (a tautology), exactly because not(|A| < |B|) can't be but (|A| = |B|) in case of |A| <= |B| proposition.


Originally Posted by doronshadmi View Post
According to you, these properties can't establish the ZF(C) Axiom of infinity to actually be the ZF(C) Axiom of infinity unless more ZF(C) axiom are involved.
Originally Posted by jsfisher View Post
Nope, I never said that. (And don't equate the name give to an axiom with what the axiom actually says. The Axiom of Infinity postulates the existence of a set with two properties. It does not call it an infinite set. Even if it had, that would not define what infinite set meant, just postulate the existence of one example.)
Originally Posted by jsfisher View Post
The Axiom must to be coupled with other axioms to conclude von Neumann's ordinal is a set in ZF.
Originally Posted by jsfisher View Post
Nope. You just need something that defines what "infinite set" means. "A set Q is infinite if and only if...."
Cardinality is a measure of the number of members of set A (notated as |A|)

Set A is called finite iff given any n in N, |A| is any particular n

Set A is called non-finite 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 "give any" holds for both finite and non-finite sets, where "for all" holds only for finite sets.

Non-finite sets have immediate or non-immediate successors exactly because given any n in N, |A| is not any particular n.

This is not the case with finite sets, they do not have immediate or non-immediate successors exactly because given any n in N, |A| is any particular n.


Mod InfoThread continued from here. You can quote or reply to any post from that or previous parts.
Posted By:zooterkin
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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 zooterkin; 23rd June 2020 at 01:42 PM.
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Old 21st June 2020, 08:42 AM   #2
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Originally Posted by doronshadmi View Post
The focus is only on |A| <= |B| proposition, which can be written also as (|A| < |B|) OR not(|A| < |B|) (a tautology), exactly because not(|A| < |B|) can't be but (|A| = |B|) in case of |A| <= |B| proposition.
No. I have defined what the expression, |A| <= |B|, means. It means there exists an injection from A to B.

I even provided meanings for |A| = |B| and for |A| < |B|.

You can, in fact, use those meanings to show that the expression |A| <= |B| is identical to (|A| < |B|) OR (|A| = |B|), but not the nonsense you produced.

The negation of |A| < |B| is definitely not |A| = |B|.
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Old 21st June 2020, 09:02 AM   #3
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Originally Posted by doronshadmi View Post
Cardinality is a measure of the number of members of set A (notated as |A|)
Fine. You've established a general notion and a notation.

Quote:
Set A is called finite iff given any n in N, |A| is any particular n
This does not define what |A| means. Instead, it assumes it is already defined so it can enter into a comparison.

Set A is called non-finite iff given any n in N, |A| is not any particular n[/quote]

Ditto.

So, still no definition for cardinality.


Quote:
...but not in my framework...
You have been blathering on about how conventional Mathematics mistreats infinity. You don't get to now take shelter in the fantastic realm of doronetics. You do not get to substitute your sometimes bizarre misconceptions to then "prove" Mathematics wrong.

But since you have, your whole argument has collapsed. You have nothing. Your criticisms of Mathematics vanish into a dust cloud of things you just make up.
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Old 22nd June 2020, 02:50 AM   #4
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Originally Posted by jsfisher View Post
No. I have defined what the expression, |A| <= |B|, means. It means there exists an injection from A to B.

I even provided meanings for |A| = |B| and for |A| < |B|.

You can, in fact, use those meanings to show that the expression |A| <= |B| is identical to (|A| < |B|) OR (|A| = |B|), but not the nonsense you produced.

The negation of |A| < |B| is definitely not |A| = |B|.
Originally Posted by jsfisher View Post
I will provide you a definition of cardinality for the purposes of this discussion.
Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)
Note that this definition requires only the introduction of mappings into the set theory.
I re-examined our last discussions, and we are not "in the same page" about the meaning of "a relative measure of "sizes" of two sets".

This is "my page":

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).

--------------------------------------------

Let _ be a placeholder for any given set.

Definition 1: Cardinality is the measure of the number of members of _ (notated as |_|)

Definition 2: Cardinality between the two sets A and B is a relative measure of their numbers of members where ( ((|A| < |B|) iff (|B| > |A|)) OR ((|A| = |B|) iff (|B| = |A|)) ) (The meanings for equality (=) and inequalities (< or >) of cardinalities follow directly (also 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)).

In simpler words, Cardinality is the measure of the number of members of sets A and B, where their possible relations are = or < or >.

It says nothing about being finite or non-finite set.

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 non-finite 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 "give any" holds for both finite and non-finite sets, where "for all" holds only for finite sets.

Non-finite sets have immediate or non-immediate successors exactly because given any n in N, |A| is not any particular n.

This is not the case with finite sets, they do not have immediate or non-immediate successors exactly because given any n in N, |A| is any particular n.
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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; 22nd June 2020 at 04:33 AM.
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Old 22nd June 2020, 05:16 AM   #5
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Originally Posted by doronshadmi View Post
Definition 2: Cardinality between the two sets A and B is a relative measure of their numbers of members where ( ((|A| < |B|) iff (|B| > |A|)) OR ((|A| = |B|) iff (|B| = |A|)) )
Your definition of cardinality depends on the definition of cardinality.
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Old 22nd June 2020, 05:58 AM   #6
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Originally Posted by jsfisher View Post
Your definition of cardinality depends on the definition of cardinality.
Definition 2 is not my definition of Cardinality, but uses Definition 1 in order to define Cardinality between two sets.

Thank you. I forgot to add the following in my previous post:


"In simpler words, Cardinality between the two sets is the relative measure of the numbers of members of sets A and B, where their possible relations are = or < or >."
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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.

Last edited by doronshadmi; 22nd June 2020 at 06:23 AM.
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Old 22nd June 2020, 06:34 AM   #7
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Yes, but

Originally Posted by jsfisher View Post
No. I have defined what the expression, |A| <= |B|, means. It means there exists an injection from A to B.

I even provided meanings for |A| = |B| and for |A| < |B|.

You can, in fact, use those meanings to show that the expression |A| <= |B| is identical to (|A| < |B|) OR (|A| = |B|), but not the nonsense you produced.

The negation of |A| < |B| is definitely not |A| = |B|.
You have still not addressed palindromic numbers as aleph bet. Non-cardinality is the merest ish kabibble in your precious set > non-set ~ und so weiter.

Why haven't you?
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Old 22nd June 2020, 09:12 AM   #8
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Originally Posted by doronshadmi View Post
Definition 2 is not my definition of Cardinality, but uses Definition 1 in order to define Cardinality between two sets.
Neither defines cardinality.

Saying "Cardinality is the measure of the number of members of [a set]" fails as a definition. It provides no information about how the measure is taken. "Number of members" is not a concept to be found anywhere in the set theory axioms.
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Old 22nd June 2020, 09:24 AM   #9
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Originally Posted by sackett View Post
You have still not addressed palindromic numbers as aleph bet. Non-cardinality is the merest ish kabibble in your precious set > non-set ~ und so weiter.

Why haven't you?
When you address palindromic numbers, they prefer to be addressed as Sir aleph bet. (Trust me, you don't want to make that mistake twice.) Last I checked, none are in any religious hierarchy, so cardinality just doesn't come up.
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Old 23rd June 2020, 07:47 AM   #10
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Originally Posted by jsfisher View Post
Saying "Cardinality is the measure of the number of members of [a set]" fails as a definition. It provides no information about how the measure is taken. "Number of members" is not a concept to be found anywhere in the set theory axioms.
Cardinality (by Definition 1) is defined as the measure of what-is-measured, where what-is-measured is "the number of members of _ (where _ is a placeholder for any given set (and it is (Cardinality) notated as |_|).

The information of how the measure is taken from what-is-measured, is provided by Definition 2.

--------------------------------------------------

Let's look at the standard definition of Cardinality:

Cardinality is a relative measure of "size" of sets where |A| <= |B| iff there exists an injection from A to B."


By the standard definition it is informed that A or B are stand for sets.

Can you tell me please what |A| or |B| are stand for, by the standard definition?
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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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Old 23rd June 2020, 09:18 AM   #11
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Originally Posted by jsfisher View Post
When you address palindromic numbers, they prefer to be addressed as Sir aleph bet. (Trust me, you don't want to make that mistake twice.) Last I checked, none are in any religious hierarchy, so cardinality just doesn't come up.
Cardinality? I thought we were talking about CARNALITY!

I

I

I feel cheated.
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Old 23rd June 2020, 09:31 AM   #12
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Originally Posted by doronshadmi View Post
Cardinality (by Definition 1) is defined as the measure of what-is-measured, where what-is-measured is "the number of members of _ (where _ is a placeholder for any given set (and it is (Cardinality) notated as |_|).

The information of how the measure is taken from what-is-measured, is provided by Definition 2.
"A measure of what-is-measured" -- What a bizarre thing to write. Meaningless and bizarre, both.

"the number of members of [a set]" -- not bizarre, but also not part of a definition. The axioms of set theory have no concept of "number of". You'd need to add that.

In the so-called Definition 2, you use three undefined relationships (<, =, >). They have no meaning in set theory unless you define them. I can only assume you are assuming a meaning from arithmetic. If so, then all you've done is implied |A| has a numeric value and shown a couple of trivial properties of the relations.
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Old 23rd June 2020, 09:33 AM   #13
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Originally Posted by sackett View Post
Cardinality? I thought we were talking about CARNALITY!

I

I

I feel cheated.
Don't give up just yet. Sir aleph bet may have carnal interests to your liking. Probably injective at the very least.
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Old 23rd June 2020, 10:38 PM   #14
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Originally Posted by jsfisher View Post
"A measure of what-is-measured" -- What a bizarre thing to write. Meaningless and bizarre, both.

"the number of members of [a set]" -- not bizarre, but also not part of a definition. The axioms of set theory have no concept of "number of". You'd need to add that.

In the so-called Definition 2, you use three undefined relationships (<, =, >). They have no meaning in set theory unless you define them. I can only assume you are assuming a meaning from arithmetic. If so, then all you've done is implied |A| has a numeric value and shown a couple of trivial properties of the relations.
Please this time answer to the following:

Let's look at the standard definition of Cardinality:

Cardinality is a relative measure of "size" of sets where |A| <= |B| iff there exists an injection from A to B."


By the standard definition it is informed that A or B stand for sets.

Can you tell me please what |A| or |B| are stand for, by the standard definition?
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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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Old 24th June 2020, 04:07 AM   #15
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Let _ be a placeholder for any given set.

Now, some corrections of definitions 1 and 2.

Definition 1: Cardinality is the "size" of _ (notated as |_|), which is determined by the members of N, where 0 is a member of N.

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).

Let < be less than.

Let > be greater than.

Let = be equal to.

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 non-finite 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 non-finite 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 non-finite sets, where "for all" holds only for finite sets.

Non-finite sets have immediate or non-immediate successors exactly because given any n in N, |A| is not any particular n.

This is not the case with finite sets, they do not have immediate or non-immediate successors exactly because given any n in N, |A| is any particular n.


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 non-strict inequality iff ((|A| ≤ |B|) iff (|B| ≥ |A|)).
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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.

Last edited by doronshadmi; 24th June 2020 at 05:04 AM.
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Old 24th June 2020, 05:03 AM   #16
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Originally Posted by doronshadmi View Post
Please this time answer to the following:

Let's look at the standard definition of Cardinality:

Cardinality is a relative measure of "size" of sets where |A| <= |B| iff there exists an injection from A to B."
The definition begins after the word, 'where'.

A and B are arbitrary sets (as was strongly implied), and |.| is a standard notation for cardinality (as was strongly implied).
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Old 24th June 2020, 05:05 AM   #17
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Originally Posted by doronshadmi View Post
Let _ be a placeholder for any given set....
Previous objects still stand. Repeating yourself instead of addressing the objections does not advance your case.
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Old 24th June 2020, 05:14 AM   #18
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Originally Posted by jsfisher View Post
Repeating yourself ...
I do not repeating my self in http://www.internationalskeptics.com...9&postcount=15 since it is a correction of http://www.internationalskeptics.com...03&postcount=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.
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Old 24th June 2020, 05:28 AM   #19
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Originally Posted by jsfisher View Post
The definition begins after the word, 'where'.

A and B are arbitrary sets (as was strongly implied), and |.| is a standard notation for cardinality (as was strongly implied).
So your definition of cardinality uses cardinality as a part of its definition, isn't it?
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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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Old 24th June 2020, 11:25 AM   #20
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Originally Posted by doronshadmi View Post
So your definition of cardinality uses cardinality as a part of its definition, isn't it?
No.

My definition (which is for a relative measure, i.e., a comparison) is:

|A| <= |B| <the thing being defined> if and only if <equivalent to saying "is defined as" or "means"> there is an injection from A to B.
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Old 25th June 2020, 06:04 AM   #21
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Originally Posted by jsfisher View Post
A and B are arbitrary sets (as was strongly implied), and |.| is a standard notation for cardinality (as was strongly implied).
If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Cardinality (which is for a relative measure, i.e., a comparison) is:

Definition of Cardinality: The cardinality of A <also notated as |A|> <= the cardinality of B <also notated as |B|> iff there is an injection from A to B.

(Also you skipped on http://www.internationalskeptics.com...8&postcount=18).
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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.

Last edited by doronshadmi; 25th June 2020 at 06:30 AM.
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Old 25th June 2020, 10:38 AM   #22
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Originally Posted by doronshadmi View Post
If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Cardinality (which is for a relative measure, i.e., a comparison) is:

Definition of Cardinality: The cardinality of A <also notated as |A|> <= the cardinality of B <also notated as |B|> iff there is an injection from A to B.
No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, |A| <= |B| is what is being defined. Not |A| nor |B|.

That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare.
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Old 28th June 2020, 05:09 AM   #23
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Originally Posted by jsfisher View Post
No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, |A| <= |B| is what is being defined. Not |A| nor |B|.

That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare.
A relative measure is defined iff what is compared and how it is compared, are both or neither taken.

In this case what is compared is the cardinalities of sets A and B (which are |A| and |B|), where how they are compared is done by one-to-one function (also called injection, such that |A| <= |B|).

Now look at this:

If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Injection (which is for a relative measure, i.e., a comparison) is:

Definition of Injection : There is a one-to-one function, also called injection, from A to B iff The cardinality of A <also notated as |A|> <= the cardinality of B <also notated as |B|>

Definition of Cardinality: The cardinality of A <also notated as |A|> <= the cardinality of B <also notated as |B|> iff there is an injection from A to B.

--------------------------------------------------------------------------------------------

(Also you skipped on http://www.internationalskeptics.com...8&postcount=18).
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Old 28th June 2020, 12:52 PM   #24
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Originally Posted by doronshadmi View Post
If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Injection (which is for a relative measure, i.e., a comparison) is....

I did not define injection. I defined a relationship between the cardinalities of two sets.
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Old 28th June 2020, 07:26 PM   #25
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Originally Posted by doronshadmi View Post
Not much of a correction, and you didn't address the objections raised.

Nonetheless...
Originally Posted by doronshadmi View Post
Let _ be a placeholder for any given set.
Or we could just follow convention and use a letter of the alphabet instead of a cryptic underbar.

Quote:
Definition 1: Cardinality is the "size" of _ (notated as |_|), which is determined by the members of N, where 0 is a member of N.
Determined how?
N? What is N? A set, I presume, but what set?
0? What is 0? Set theory has nothing like that. You'll need to define it.

Quote:
Let < be less than.
"Less than"? How is that defined? Set theory doesn't have a less-than concept on its own.

...and so on for the rest of the "corrected" post.


You have not moved any closer to a definition. Please keep trying, though.
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Old 29th June 2020, 03:37 AM   #26
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Originally Posted by jsfisher View Post
I did not define injection. I defined a relationship between the cardinalities of two sets.
In other words, you have missed what is written in http://www.internationalskeptics.com...1&postcount=23.

Please read all of it, then think about it, and only then please air your view about its content.

Thank you.
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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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Old 29th June 2020, 03:48 AM   #27
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Originally Posted by jsfisher View Post
Not much of a correction, and you didn't address the objections raised.

Nonetheless...


Or we could just follow convention and use a letter of the alphabet instead of a cryptic underbar.



Determined how?
N? What is N? A set, I presume, but what set?
0? What is 0? Set theory has nothing like that. You'll need to define it.



"Less than"? How is that defined? Set theory doesn't have a less-than concept on its own.

...and so on for the rest of the "corrected" post.


You have not moved any closer to a definition. Please keep trying, though.
As long as you try to understand http://www.internationalskeptics.com...9&postcount=15 in terms of ZF(C), you are surly missing it.
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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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Old 29th June 2020, 06:00 AM   #28
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Originally Posted by doronshadmi View Post
In other words, you have missed what is written in ...
No matter how many times one might read it, it is still gibberish.
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Old 29th June 2020, 06:07 AM   #29
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Originally Posted by doronshadmi View Post
As long as you try to understand http://www.internationalskeptics.com...9&postcount=15 in terms of ZF(C), you are surly missing it.
That post defines understanding at many levels and in so many ways. Be that as it may, though, you are the one trying to discredit set theory. That means you are the one who needs to work in terms of set theory.

...but you cannot, can you?
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Old 30th June 2020, 06:24 AM   #30
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Originally Posted by jsfisher View Post
No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, |A| <= |B| is what is being defined. Not |A| nor |B|.

That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare.
jsfisher, no relative measure is defined unless what is compared (and in this case it is the cardinality of set A (notated as |A|) and the cardinality of set B (notated as |B|)) and how it is compared (and in this case it is an injection from set A to set B), where the logical connective between what is compared and how it is compared is iff, otherwise |A| <= |B| can't be defined.

Any attempt to define X in terms of relative measure, can't avoid X as a part of the definition.

And this is exactly what I show in http://www.internationalskeptics.com...1&postcount=23, which you get as gibberish as long as you are missing the truth table of iff where:

In case of relative measure

Code:
 
p="what is measured" 
q="how it is muasured"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T
where relative measure can't be established unless p iff q.
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That is also over the matrix, is aware of the matrix.

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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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Old 30th June 2020, 07:28 AM   #31
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Originally Posted by jsfisher View Post
That post defines understanding at many levels and in so many ways.
What is defined in http://www.internationalskeptics.com...9&postcount=15 is simpler than your definitions that are done in terms of ZF(C) (your definition of Cardinality, which uses cardinality as a part of the definition).

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 non-finite (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 non-strict inequality as a range between strict inequality (< or >) and equality (=).

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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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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Old 30th June 2020, 10:03 AM   #32
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Originally Posted by doronshadmi View Post
jsfisher, no relative measure is defined unless what is compared (and in this case it is the cardinality of set A (notated as |A|) and the cardinality of set B (notated as |B|)) and how it is compared (and in this case it is an injection from set A to set B), where the logical connective between what is compared and how it is compared is iff, otherwise |A| <= |B| can't be defined.

You used a lot of words there. Did you have a point?

My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.

...and if you can't get along without supplemental information, no matter how obvious: ...where A and B are each sets, |.| is used to denote "cardinality of", and <= is the relationship that is the subject of the definition.
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Old 30th June 2020, 10:06 AM   #33
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Originally Posted by doronshadmi View Post
Originally Posted by jsfisher View Post
That post defines understanding at many levels and in so many ways.
What is defined in ....

My post contained a typographic error. The word intended was 'defies'. I apologize for the confusion.
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Old 1st July 2020, 03:09 AM   #34
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Originally Posted by jsfisher View Post
My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.
Let's see:

Originally Posted by jsfisher View Post
My definition (which is for a relative measure, i.e., a comparison) is:

|A| <= |B| <the thing being defined> if and only if <equivalent to saying "is defined as" or "means"> there is an injection from A to B.
So, your definition is for:

Relative measure.

Relative cardinalities.

Will you make up your mind?

Originally Posted by jsfisher View Post
Here is your first post about it:

For that matter, you need to be clear what you mean by "cardinality". No, wait, better still, I will provide you a definition of cardinality for the purposes of this discussion.

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)

Note that this definition requires only the introduction of mappings into the set theory
Mapping between sets' cardinalities, so you are using cardinality as a part of your definition of Cardinality.

Moreover, you have said that you will provide me a definition of cardinality.

Originally Posted by jsfisher View Post
...and if you can't get along without supplemental information, no matter how obvious:
Obvious?
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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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Old 1st July 2020, 08:23 AM   #35
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Originally Posted by doronshadmi View Post
Let's see:



So, your definition is for:

Relative measure.

Relative cardinalities.

Will you make up your mind?

That' it? Petty attempts at semantic nitpicks?

Come back when you have something of substance. Meanwhile, Mathematics remains intact.
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Old 1st July 2020, 07:25 PM   #36
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Originally Posted by doronshadmi View Post
Let's see:



So, your definition is for:

Relative measure.

Relative cardinalities.

Will you make up your mind?



Mapping between sets' cardinalities, so you are using cardinality as a part of your definition of Cardinality.

Moreover, you have said that you will provide me a definition of cardinality.



Obvious?
Once again, you get things wrong. jsfisher is NOT "mapping between set's cardinality" so he is NOT using cardinality as a part of his definition. Moreover, is has provided you a definition of cardinality, multiple times.

Post #2: No. I have defined what the expression, |A| <= |B|, means. It means there exists an injection from A to B.
Post #4: Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (quoted by you!)
Post #14: Cardinality is a relative measure of "size" of sets where |A| <= |B| iff there exists an injection from A to B. (quoted by you, again)
Post #20: |A| <= |B| <the thing being defined> if and only if <equivalent to saying "is defined as" or "means"> there is an injection from A to B.
Post #21 is you misquoting (on purpose?) the definition given in post #20.
Post #23 is again you misquoting (on purpose?) the definition given in post #20.
Post #32: My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.
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Old 1st July 2020, 11:45 PM   #37
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Originally Posted by Little 10 Toes View Post
jsfisher is NOT "mapping between set's cardinality" so he is NOT using cardinality as a part of his definition.
By jsfisher's definition of Cardinality, |A| and |B| (which are actually the cardinalities of sets A and B) are inseparable parts of his definition, exactly as injection function from set A to set B is an inseparable part of his definition ,where this inseparability is based on IFF logical connective as as seen by its truth table, about the considered case, which is:

p="the cardinality of set A (notated as |A|) <= the cardinality of set B (notated as |B|))"

q= "there is injection from set A to set B (notated as <= between the cardinality of set A (notated as |A|) and the cardinality of set B (notated as |B|)"

Here is the iff truth table

Code:
 
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T
and as logically seen, cardinality is an inseparable part of his definition of Cardinality, such that p and q are both true or both false.

Originally Posted by Little 10 Toes View Post
Moreover, is has provided you a definition of cardinality, multiple times.
And in all these times, cardinality is an inseparable part of his definition of Cardinality.

-------------------

Also please see my post about relative measure in http://www.internationalskeptics.com...7&postcount=30
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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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Old 2nd July 2020, 12:34 AM   #38
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Originally Posted by jsfisher View Post
That' it? Petty attempts at semantic nitpicks?

Come back when you have something of substance. Meanwhile, Mathematics remains intact.
Meanwhile cardinality is an inseparable part of your definition of Cardinality (as shown in http://www.internationalskeptics.com...9&postcount=37) where this "definition" is a fundamental piece of your mathematical framework about sets.
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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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Old 2nd July 2020, 02:07 AM   #39
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Originally Posted by jsfisher View Post
That' it? Petty attempts at semantic nitpicks?
Who is the one that provides those semantics as parts of his posts about the definition of Cardinality? (the answer is given in http://www.internationalskeptics.com...5&postcount=34)
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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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Old 2nd July 2020, 03:30 AM   #40
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Originally Posted by jsfisher View Post
My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.

...and if you can't get along without supplemental information, no matter how obvious: ...where A and B are each sets, |.| is used to denote "cardinality of", and <= is the relationship that is the subject of the definition.
Ok let's re-examine what you wrote as follows:

An injection from set A to set B is a one-to-one function from the members of set A to the members of set B, such that for every member of set A there is at most one matched member from set B, which provides at most two options by this relative measurement as follows:

(The "size" of set A is relatively less than the size of set B) OR (The "size" of set A is relatively the same as the "size" of set B).

Now comes the trick of "washing-machine of words to symbols" by replacing "the "size" of set A" with |A| symbol, and "the "size" of set B" with symbol |B|.

Also by this trick of "washing-machine of words to symbols" "relatively less than" is replaced by <, and "relatively the same as the "size" of" is replaced by symbol =.

And now after the "washing-machine of words to symbols" did its job, we can write:

(|A|<|B|) OR (|A|=|B|) (also written as |A|<=|B|) means (written as iff logical connective) that there is injection from set A to set B.

Now we take this "washing-machine of words to symbols" result and express it like this:

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)

------------------------------------------------------------

jsfisher, no matter how your "washing-machine of words to symbols" works, you are using cardinality as a part of your definition of Cardinality.

------------------------------------------------------------

A more straightforward framework of this subject can be seen in http://www.internationalskeptics.com...8&postcount=31, where any criticism is welcome.

Originally Posted by jsfisher View Post
That' it? Petty attempts at semantic nitpicks?

Come back when you have something of substance. Meanwhile, Mathematics remains intact.
Mathematics remains intact according to traditional mathematicians, if they take collections and their relations in terms of Plato philosophy school of thought.

jsfisher, since you are using the term "intact" I believe that "set N is a complete whole" may be understood by you as "set N is intact".
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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.

Last edited by doronshadmi; 2nd July 2020 at 04:24 AM.
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