Cont: Deeper than primes - Continuation 2

[jsfisher]

https://en.wikipedia.org/wiki/Cardinality

How nice! You quote a section from an article on cardinality suitable for elementary school students.

You did not show that "The meanings for strict equality and strict inequality of cardinalities follow directly" from the definition of Cardinality, which "is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B" since you did define strict or relative measure
So please define them before we continue.

Are you not familiar with the my usage of the word, strict, with respect to comparisons? The < and > relationships are the strict inequalities since the exclude the possibility of equality. Similarly, = is a strict equality relationship since it excludes the possibility of inequality.

This is all standard stuff.

As for 'relative measure', I have stated that cardinality is a measure (you understand that term, right?) of the "size" of a set. (Note the use of quotation marks. Just like in Wikipedia's "number of elements", the usage is figurative, not literal.) Measures can be relative if they are defined by a relationship. |A| <= |B| simply states that the size of A is less than or equal to the size of B. This is about as primitive meaning for cardinality you can have, and it only requires the set theory to include the concept of mappings. Other equivalent meanings for cardinality can be derived from this, but not without considerably more framework being added to the underlying set theory.

Doronshadmi, it helps not at all if you do not understand the basic vocabulary, and you only make it worse by introducing your own, special, non-standard terms and phrases without even a vague attempt to define them.

Be that as it may...

|A| <= |B| iff there exists an injection from A to B.
|A| = |B| iff |A| <= |B| and |B| <= |A|
|A| < |B| iff |A| <= |B| and not |B| <= |A|​

 

[jsfisher]

jsfisher did not address it.

He used "strict" and "relative measure" and also claimed about "strict equality and strict inequality of cardinalities that follow directly form "a relative measure of "size" of sets", without defining "strict" or "a relative measure of "size" of sets".

Nor should I have had to.

Moreover, by looking at wikipadia, "In mathematics, the cardinality of a set is a measure of the "number of elements" of the set." where the word "relative" is not used.

You might want to peak just past the introductory paragraph intended for a third-graders...unless you like all that egg on your face.

 

[doronshadmi]

How nice! You quote a section from an article on cardinality suitable for elementary school students.

Being arrogant does not help to support your arguments.

|A| <= |B| simply states that the size of A is less than or equal to the size of B.

Let's write down the Axiom of infinity (https://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement):

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

∃ I ( ∅ ∈ I ∧ ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) )

In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

Inductive sets have inherently next objects by this axiom (for any x ∈ I there is inherently the next ( ( x ∪ { x } ) ∈ I ) ), therefore their strict size is undefined (the quantifier ∀ (for all) can't be used in case of deductive sets).


Definition 1: Sets have strict sizes iff no next objects inherently related to them.

Definition 2: |A| is strictly less than |B| OR |A| is strictly equal to |B| iff |A| and |B| are strict sizes.

By definitions 1 and 2 only finite sets do not have inherently next objects.

Definition 3: The order of the members of a given set does not change its size.

So, whether there is always at least one next c number between any two given a<b numbers (where a and b are not taken only as integers), or given any I member, there is inherently at least one next member, in both cases infinite sets do not have strict sizes.

 

[jsfisher]

Being arrogant does not help to rest your case.

It wasn't arrogance. It was an observation based in fact.

Let's write down the Axiom of infinity...

You really should credit Wikipedia for that excerpt. Be that as it may, yes, we are familiar with the axiom.

Inductive sets have inherently next objects by this axiom

Next objects? What does that mean? The axiom states there exists a set that has two properties. It doesn't claim anything else, and it especially doesn't claim anything about ordering, so how does "next" come into this?

...therefore their strict size is undefined.

By "size", are you referring to cardinality? This conclusion certainly doesn't follow from my definition for cardinality, so what's yours?

At this point, you've hit your limit for undefined things, so ...<snip>...

 

[jsfisher]

It is probably time to make a list of words and phrases with meanings unknown to the rest of us.

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set
• complete whole
• complete set

Not that I expect Doronshadmi to address any of them, I suspect his personal definition for cardinality would provide the most insight into his confusion.

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set
• complete whole
• complete set
• next object

 

[doronshadmi]

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set
• complete whole
• complete set
• next object

Please read very carefully http://www.internationalskeptics.com/forums/showpost.php?p=13124393&postcount=3463 once again.

 

[jsfisher]

Please read very carefully http://www.internationalskeptics.com/forums/showpost.php?p=13124393&postcount=3463 once again.

Why?

I do see that you changed it from the time I commented on it. You removed a paragraph about some examples. Just as well since the paragraph was entirely wrong, but you now have a final sentence that sits orphaned from any context.

In context, it was completely wrong. Now, it just sits there as gross confusion.


Back to the vocabulary list: How do you define cardinality?

 

[Hevneren]

Definition 1: Sets have strict sizes iff no next objects inherently related to them.

I can't parse this sentence.

Definition 2: |A| is strictly less than |B| OR |A| is strictly equal to |B| iff |A| and |B| are strict sizes.

This doesn't seem to make sense. So |B| is never strictly less than |A|, if |A| and |B| are strict sizes? What if I choose to swap the letters A and B?

By definitions 1 and 2 only finite sets do not have inherently next objects.

Definition 3: The order of the members of a given set does not change its size.

I can't see anything being defined here

So, whether there is always at least one next c number between any two given a<b numbers (where a and b are not taken only as integers), or given any I member, there is inherently at least one next member, in both cases infinite sets do not have strict sizes.

There are currently 7.8 billion people in this world. Only one of them would "come to" such a "conclusion".

 

[doronshadmi]

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.

Measures can be relative if they are defined by a relationship. |A| <= |B| simply states that the size of A is less than or equal to the size of B. This is about as primitive meaning for cardinality you can have, and it only requires the set theory to include the concept of mappings.

Ok let's examine your primitive meaning for cardinality, by defining the cardinality of a given set.

In that case the relationship is between the same members (the relative measure of the given set is on itself (the domain and the codomain have the same members)).

------------------------------
Definition 1: Changing the order of the members of a given set, does not change its cardinality.

Example (in case of a finite set): |{1,2,3,4,5}| = |{5,3,1,4,2}|

Example (in case of an infinite set): |{1,2,3,4,5,...}| = |{5,3,1,4,2,...}|
------------------------------

Now, let's write down the Axiom of infinity (https://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement):

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

∃ I ( ∅ ∈ I ∧ ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) )

In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

So, by this axiom for any x ∈ I, x implies ( ( x ∪ { x } ) ∈ I ) ), therefore (by definition 1 and by the shown logical implication) I cardinality is undefined (which means that the quantifier ∀ (for all) can't be used in case of inductive sets).

An example:

The set of natural numbers is some case of an inductive set, and by definition 1 and by the logical implication, shown by the Axiom Of Infinity, it is shown that the cardinality of the set of natural numbers is undefined, since the biggest member of the set of natural numbers does not exist.

 

[doronshadmi]

So, the best that can be provided by

∃ I ( ∅ ∈ I ∧ ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) )

is no more than an endless logical implication, such that x ∈ I implies ( ( x ∪ { x } ) ∈ I ) only finitely many times, where:

1) The biggest result of x ∪ { x } does not exist.

2) The result of x ∪ { x } is always finitely long.

3) By (1) and (2) the cardinality of I is always finite but undefined, unlike sets that their cardinality is finite but defined (for example: |{}| , |{{}}| , |{{},{{}}}| , etc.)

Finite but undefined is actually the property of the cardinality of potential infinity.

 

[jsfisher]

Ok let's examine your primitive meaning for cardinality, by defining the cardinality of a given set.

I already defined cardinality.

In that case the relationship is between the same members (the relative measure of the given set is on itself (the domain and the codomain have the same members)).

No. The relationship is between any two sets.

------------------------------
Definition 1: Changing the order of the members of a given set, does not change its cardinality.

Example (in case of a finite set): |{1,2,3,4,5}| = |{5,3,1,4,2}|

Example (in case of an infinite set): |{1,2,3,4,5,...}| = |{5,3,1,4,2,...}|

This is meaningless since it refers to a property sets do not have. Moreover, even if the property did exist for sets, this would not be a definition, a lemma perhaps, but certainly not a definition.

No point in going any further with your post, Doronshadmi, until you address these above-cited blunders.

 

[Little 10 Toes]

It's so funny to see Doron Shadmi work himself into a lather and get nowhere. He can't even define cardinality.

 

[jsfisher]

It is probably time to make a list of words and phrases with meanings unknown to the rest of us.

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set
• complete whole
• complete set

Not that I expect Doronshadmi to address any of them, I suspect his personal definition for cardinality would provide the most insight into his confusion.

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set
• complete whole
• complete set
• next object

I need to add another:

• Biggest result

 

[jsfisher]

It's so funny to see Doron Shadmi work himself into a lather and get nowhere. He can't even define cardinality.

Its humor value is this thread's only saving grace.

I think Doronshadmi's inability to define anything, not just cardinality, is because he is limited to the elementary school framework of arithmetic. Don't know for sure if I am right on that, but it would explain many things.

 

[doronshadmi]

Ok let's examine your primitive meaning for cardinality, by defining the cardinality of a given set.

I already defined cardinality

Sorry for the confusion, but by "defining the cardinality of a given set" I mean that I use your definition of Cardinality in case of |A| = |B| (where |A| = |B| iff (|A| < OR = |B|) AND (|B| < OR = |A|)).

It can be done since OR is involved in the definition of Cardinality ( "<=" is "< OR ="), so one of the possible cases is to find the cardinality of a given set by measure its "size" relatively to itself (no two sets are needed in order to find the cardinality of a given set).

Now, let's write down the Axiom of infinity (https://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement):

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

∃ I ( ∅ ∈ I ∧ ∀ x ∈ I ( ( x ∪ { x } ) ∈ I ) )

In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

So, by this axiom for any x ∈ I, x implies ( ( x ∪ { x } ) ∈ I ) ), therefore by the shown logical implication, I cardinality is undefined (which means that the quantifier ∀ (for all) can't be used in case of inductive sets).

An example:

The set of natural numbers is some case of an inductive set and by the logical implication of the Axiom Of Infinity, it is shown that the cardinality of the set of natural numbers is undefined, since the biggest member of the set of natural numbers does not exist.

 

[jsfisher]

Sorry for the confusion, but by "defining the cardinality of a given set" I mean that I use your definition of Cardinality in case of |A| = |B| (where |A| = |B| iff (|A| < OR = |B|) AND (|B| < OR = |A|)).

It can be done since OR is involved in the definition of Cardinality ( "<=" is "< OR ="), so one of the possible cases is to find the cardinality of a given set by measure its "size" relatively to itself (no two sets are needed in order to find the cardinality of a given set).

The cardinality of a set is equal to the cardinality of that set. For all sets A, |A| = |A|. That is all you can conclude for the meaning of cardinality as defined.

This seems like a trivial observation, Doronshadmi. Why do you bring it up?

Now, let's write down the Axiom of infinity....

We have all seen it before. No need to plagiarize Wikipedia yet again.

So, by this axiom for any all x ∈ I, x implies ( ( x ∪ { x } ) ∈ I ) )

Fixed it for you. Don't ad lib. Stick to what the axiom says. Your "x implies...", in particular, shouldn't be there.

therefore by the shown logical implication, I cardinality is undefined

Sorry, but where precisely was that shown? How, exactly, do you link ∀x ∈ I ((x ∪ {x}) ∈ I) to cardinality? Not seeing the connection.

(which means that the quantifier ∀ (for all) can't be used in case of inductive sets).

The rules for predicate calculus would disagree.

An example:

The set of natural numbers is some case of an inductive set and by the logical implication of the Axiom Of Infinity, it is shown that the cardinality of the set of natural numbers is undefined, since the biggest member of the set of natural numbers does not exist.

Well, you didn't show any of that. Besides, you do realize that there are infinite sets that are not inductive sets, right? And there are infinite sets that do have a "biggest member" (under some reasonable definition of biggest), right?

So, not only did you not show the cardinality of the set of natural numbers to be undefined (in fact, the very definition of cardinality you agreed to at the start of your post makes that clear), your conclusion is unconnected to inductive sets or "biggest members".

 

[Little 10 Toes]

Is it just me or is there the easy way and then there's the DoronShadmi way?

 

[doronshadmi]

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

Let's examine this part: "where |A| <= |B| if and only if there exists an injection from A to B."

So, by your definition of cardinality A is a domain and B is a codomain.

1) The notation |A| <= |B| means that |A| is less than or equal to |B| (or, equivalently, at most b, or not greater than b).

2) In case of injection, since every distinct member of A is mapped exactly to one distinct member of B AND there are distinct members of B that no distinct one member of A is mapped to them, |A| can't be at most |B|, or |A| not greater than |B|, which means that if there exists an injection from A to B only |A|<|B| holds (since injection is not the same as bijection, and you used only injection from |A| to |B| in your definition of Cardinality).

3) You may claim that by writing injection you actually mean injective-only OR bijective, but in that case <= is actually non-strict inequality (see (1)) (so, the meanings for strict equality and strict inequality of cardinalities does not follow directly from your definition, as you claim).

About non-strict inequality, please see https://en.wikipedia.org/wiki/Inequality_(mathematics)

About injection, please see https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

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

Anyway, how your definition of Cardinality is related to sets with infinitely many members?

 

[jsfisher]

Let's examine this part: "where |A| <= |B| if and only if there exists an injection from A to B."

So, by your definition of cardinality A is a domain and B is a codomain.

No. There is an injection for which A (not its cardinality) is the domain and B (not its cardinality) is the codomain.

ETA: Read the original post too quickly.

1) The notation |A| <= |B| means that |A| is less than or equal to |B| (or, equivalently, at most b, or not greater than b).

2) In case of injection, since every distinct member of A is mapped exactly to one distinct member of B AND there are distinct members of B that no distinct one member of A is mapped to them...

Nope, not that last part. And as a result, everything that followed in your post was wrong.

...<snip>...

 

[jsfisher]

Anyway, how your definition of Cardinality is related to sets with infinitely many members?

Nothing in my definition for cardinality is affected by whether the sets are finite or infinite.