Cont: Deeper than primes - Continuation 1/3*9

doronshadmi

Penultimate Amazing
doronshadmi
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jsfisher said:
|A| <= |B| if and only if / is defined by / means there exists an injection from A to B​
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You wrote
jsfisher said:
Given some set A and some set B, |A| <= |B| is a proposition that may or may not be true
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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.


doronshadmi said:
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.
jsfisher said:
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.)
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jsfisher said:
The Axiom must to be coupled with other axioms to conclude von Neumann's ordinal is a set in ZF.
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jsfisher said:
Nope. You just need something that defines what "infinite set" means. "A set Q is infinite if and only if...."
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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.


Thread continued from here. You can quote or reply to any post from that or previous parts.
Posted By: zooterkin
 
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doronshadmi said:
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.
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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|.
 
doronshadmi said:
Cardinality is a measure of the number of members of set A (notated as |A|)
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Fine. You've established a general notion and a notation.

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


...but not in my framework...
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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.
 
jsfisher said:
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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jsfisher said:
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.
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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).

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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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doronshadmi said:
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|)) )
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Your definition of cardinality depends on the definition of cardinality.
 
jsfisher said:
Your definition of cardinality depends on the definition of cardinality.
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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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Yes, but

jsfisher said:
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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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?
 
doronshadmi said:
Definition 2 is not my definition of Cardinality, but uses Definition 1 in order to define Cardinality between two sets.
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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.
 
sackett said:
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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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.
 
jsfisher said:
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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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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jsfisher said:
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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Cardinality? I thought we were talking about CARNALITY!

I

I

I feel cheated.
 
doronshadmi said:
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.
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"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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sackett said:
Cardinality? I thought we were talking about CARNALITY!

I

I

I feel cheated.
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Don't give up just yet. Sir aleph bet may have carnal interests to your liking. Probably injective at the very least.
 
jsfisher said:
"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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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?
 
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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doronshadmi said:
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."
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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).
 
jsfisher said:
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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So your definition of cardinality uses cardinality as a part of its definition, isn't it?
 
doronshadmi said:
So your definition of cardinality uses cardinality as a part of its definition, isn't it?
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No.

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

|A| <= |B| [SIZE=-2]<the thing being defined>[/SIZE] if and only if [size=-2]<equivalent to saying "is defined as" or "means">[/size] there is an injection from A to B.
 
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