Cont: Deeper than primes - Continuation 2

[doronshadmi]

Not all closed curves are circles, but all circles are closed curves. Not all injections are bijections, but all bijections are injections.

"Not all closed curves are circles, but all circles are closed curves." is right, since a circle is a particular case of a closed curve.

"Not all injections are bijections, but all bijections are injections." is wrong, since
(Not even a single one-to-one is one-to-one AND onto) AND (Not even a single one-to-one AND onto is one-to-one).

My statement remains correct.

You have two statements, where one is right and one is wrong, as shown above.

 

[doronshadmi]

Why not? This is standard stuff, Doronshadmi. From any standard sort of meaning of the <= relationship, one can readily derive meanings for < and =.

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.)

You have claimed that strict equality and strict inequality of cardinalities follow directly from your definition, but your definition is wrong since
(Not even a single one-to-one is one-to-one AND onto) AND (Not even a single one-to-one AND onto is one-to-one).

Well, that's a very limited definition for cardinality. Got a definition that works for all sets?.

Please define "all sets"?

It is also clear you are building your conclusion into your definitions.

Please support your claim.

Meanwhile:

(|A| is called strict iff for bijection f : A → { 0 , … , n } |A| is some n) OR (|A| is called non-strict iff for bijection f : A → { 0 , … , n } |A| is not any some n)

Cardinality is a measure of the number of members of set A, such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).

By non-strict I mean that |A| is not any some n, since the number of members of set A is changed by successor operation.

As always, you are free to develop your whole branch of doronetics, with whatever special definitions you like, but you don't get to drag your nonsense into a discussion of normal Mathematics just to claim it is wrong.

It isn't, but you are.

"normal Mathematics" is no less no more than a philosophical view, which is agreed among a group of persons.

These persons believe that if something is defined by their agreed rules, it can't be criticized unless it is done by their agreed rules.

But the beauty of Mathematics is that the agreed rules themselves are not beyond criticism.

In this discussion I criticize the agreed rules that are used to establish fundamental concepts like Cardinality, Infinity, Mapping etc.

The most important thing is that there are no things that are beyond criticism, including me of course.

 

[jsfisher]

"Not all closed curves are circles, but all circles are closed curves." is right, since a circle is a particular case of a closed curve.

And bijections are a particular case of injections. Why is this so hard for you to understand?

 

[doronshadmi]

And bijections are a particular case of injections. Why is this so hard for you to understand?

Thank you jsfisher, you are right and I am wrog, so all of what I wrote that is based on this mistake "is thrown out of the window".

So what is left is this:

(|A| is called strict iff for bijection f : A → { 0 , … , n } |A| is some n) OR (|A| is called non-strict iff for bijection f : A → { 0 , … , n } |A| is not any some n)

Cardinality is a measure of the number of members of set A, such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).

By non-strict I mean that |A| is not any some n, since the number of members of set A is changed by successor operation.

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

The standard definition:

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.

This definition does not distinguish between finite or infinite cardinality.

In that case I'll ask you: How, for example, |N| is defined as infinite (or transfinite) cardinality, and |n| is defined as finite cardinality?

 

[jsfisher]

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.

This definition does not distinguish between finite or infinite cardinality.

No, it does not. Why should it?

If you think about it, finite and non-finite refer to size, so you need a size measure before you can define terms like finite and non-finite.

In that case I'll ask you: How, for example, |N| is defined as infinite (or transfinite) cardinality, and |n| is defined as finite cardinality?

It doesn't...yet.

Just like set theory can be developed to include functions, and the whole numbers, and arithmetic, and so on, cardinality can also be developed. I provided a simple definition for cardinality as a measure of relative size. It is the foundational definition. It will also be there. But I can build on it....

One of the first things I can add is a concept of absolute measure. Cardinality, as presented, is a comparison. It is often useful to have the measure return a "number" if you will (note the use of quotes). Rather than comparing sets with each other, I could compare them against some standard.

The von Neumann set is a candidate for such a standard. You seemed to hint at this in your attempts to define cardinality. If the cardinality of some set, S, is equal to the cardinality of a element, n, of the von Neumann set then we can say, by definition, more or less, that |S| = n.

Then, once we recognize the von Neumann set can represent the set of natural numbers, we have developed the foundational definition of cardinality into one more familiar to most non-mathematicians. "Cardinality is the number of elements in a set."

There's a problem, though. Not all sets have a cardinality equal to an element of the von Neumann set. The foundational definition still applies to all sets, but this attempt at an absolute measure works for only some of the possible sets.

Here is where we can introduce concepts of finite and non-finite sets. (Finite sets are those with cardinalities equal to an element of the von Neumann set....)

A naive attempt to include the non-finite sets into our absolute measure is to extend the von Neumann set with a new element we can call 'infinity'. Finite sets have a whole number for their cardinality, and the others all have 'infinity' for theirs.

But....as Cantor pointed out, non-finite sets come in different sizes. Our naive attempt to include the non-finite sets in our absolute measure comes up short. More extensions are required. The transfinites arise, and so on.

Note, though, that the foundational definition for cardinality has not changed. We have sought a standard that can be used for an absolute measure, and that proved to be non-trivial, but cardinality in its basic form remains as it always was.

 

[doronshadmi]

No, it does not. Why should it?

If you think about it, finite and non-finite refer to size, so you need a size measure before you can define terms like finite and non-finite.



It doesn't...yet.

Just like set theory can be developed to include functions, and the whole numbers, and arithmetic, and so on, cardinality can also be developed. I provided a simple definition for cardinality as a measure of relative size. It is the foundational definition. It will also be there. But I can build on it....

One of the first things I can add is a concept of absolute measure. Cardinality, as presented, is a comparison. It is often useful to have the measure return a "number" if you will (note the use of quotes). Rather than comparing sets with each other, I could compare them against some standard.

The von Neumann set is a candidate for such a standard. You seemed to hint at this in your attempts to define cardinality. If the cardinality of some set, S, is equal to the cardinality of a element, n, of the von Neumann set then we can say, by definition, more or less, that |S| = n.

Then, once we recognize the von Neumann set can represent the set of natural numbers, we have developed the foundational definition of cardinality into one more familiar to most non-mathematicians. "Cardinality is the number of elements in a set."

There's a problem, though. Not all sets have a cardinality equal to an element of the von Neumann set. The foundational definition still applies to all sets, but this attempt at an absolute measure works for only some of the possible sets.

Here is where we can introduce concepts of finite and non-finite sets. (Finite sets are those with cardinalities equal to an element of the von Neumann set....)

A naive attempt to include the non-finite sets into our absolute measure is to extend the von Neumann set with a new element we can call 'infinity'. Finite sets have a whole number for their cardinality, and the others all have 'infinity' for theirs.

But....as Cantor pointed out, non-finite sets come in different sizes. Our naive attempt to include the non-finite sets in our absolute measure comes up short. More extensions are required. The transfinites arise, and so on.

Note, though, that the foundational definition for cardinality has not changed. We have sought a standard that can be used for an absolute measure, and that proved to be non-trivial, but cardinality in its basic form remains as it always was.

jsfisher, thank you for providing a " simple definition for cardinality as a measure of relative size" as "the foundational definition" that you "can build on it...."

But before what I wrote is "thrown out of the window" let's very carefully recheck injection.

By injection from domain A to co-domain B |A| <= OR =< |B| (where <= is (< OR =) and =< is (= OR <)).

Let X be injection, such that X is (((X-only) OR not(X-only)) OR ((not(X-only) OR (X-only))) (where the bolded ( or ) decoration are used only for visual clarification).

In first order logic ((not for all X(X=X) OR (for all X(X=X)) is a tautology.

Here is your analogy:

"Not all closed curves are circles, but all circles are closed curves."

By your analogy "closed curves" are equivalent to (((X-only) OR not(X-only)) OR ((not(X-only) OR (X-only))) and "circles" are equivalent to NOT(((X-only) OR not(X-only)) OR ((not(X-only) OR (X-only))), but by first order logic "closed curves" are equivalent to "(not for all X(X=X))" and "circles" are equivalent to "(for all X(X=X))"

So according to your analogy:

(((X-only) OR not(X-only)) OR ((not(X-only) OR (X-only))) = (not for all X(X=X)), which is False.

OR

NOT(((X-only) OR not(X-only)) OR ((not(X-only) OR (X-only))) = (for all X(X=X)), which is False.

Since the truth value of False OR False is False I don't see how your definition of Cardinality has a logical True basis.

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

Please show my mistake.

Thank you.

 

[doronshadmi]

Here is a wikipedia paragraph that demonstrates the motivation behind the standard definition of cardinality: ( https://en.wikipedia.org/wiki/Cardinal_number#Motivation )

A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y. An injective mapping identifies each element of the set X with a unique element of the set Y. This is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size we would observe that there is a mapping:

1 → a
2 → b
3 → c

which is injective, and hence conclude that Y has cardinality greater than or equal to X. The element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injective and onto mapping. The advantage of this notion is that it can be extended to infinite sets.

If you think about it, finite and non-finite refer to size, so you need a size measure before you can define terms like finite and non-finite.

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.

A naive attempt to include the non-finite sets into our absolute measure is to extend the von Neumann set with a new element we can call 'infinity'. Finite sets have a whole number for their cardinality, and the others all have 'infinity' for theirs.

All what is written up there is very nice, but still does not provide a non-naive way to be extended into infinite sets.

So please show how the extension into infinite sets is not-naively done (by not-naively I also mean that it is not done arbitrarily).

Without it "The advantage of this notion is that it can be extended to infinite sets." or ",and the others all have 'infinity' for theirs." is no more than an assertion.

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

If you think about it, finite and non-finite refer to size, so you need a size measure before you can define terms like finite and non-finite.

If you think about it, you need the property of "size" of sets before you can define "size" of sets.

(|A| is called strict iff for bijection f : A → { 0 , … , n } |A| is some n) OR (|A| is called non-strict iff for bijection f : A → { 0 , … , n } |A| is not any some n)

Cardinality is a measure of the number of members of set A, such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).

By non-strict I mean that |A| is not any some n, since the number of members of set A is changed by successor operation.

 

[Little 10 Toes]

DoronShadmi, Please provide your definition of cardinality first. what's the point of the discussion when you do not agree on the definition.

Edit: and how about you mark what part of your posts you keep changing.

 

[jsfisher]

jsfisher, thank you for providing a " simple definition for cardinality as a measure of relative size" as "the foundational definition" that you "can build on it...."

That's it? No actual commentary on the content? Just you double-downing on your own unrelated misconceptions?

By injection from domain A to co-domain B |A| <= OR =< |B| (where <= is (< OR =) and =< is (= OR <)).

Can you express this in a well-formed way instead of this gibberish? And what is wrong with the original?

|A| <= |B| iff there is an injection from A to B.​

It does not require any embellishment from you, and certainly not embellishment that is notational nonsense.


Your post continues in this way with things that are not Mathematics. Logical conjunction, for example, does not accept functions for its operands. Please stop inventing notation without meaning which you then morph into even more meaningless notation.


Also, please stop trying to disprove definitions. They are not subject to proof.

 

[Little 10 Toes]

DoronShadmi, Please stop changing your posts after someone posts. Your constant dishonesty is showing. Again.

 

[jsfisher]

All what is written up there is very nice, but still does not provide a non-naive way to be extended into infinite sets.

Conveniently, my definition for cardinality requires no extension for infinite sets.

 

[doronshadmi]

That's it? No actual commentary on the content?

http://www.internationalskeptics.com/forums/showpost.php?p=13129316&postcount=3507

 

[doronshadmi]

Finite sets have a whole number for their cardinality, and the others all have 'infinity' for theirs.

Conveniently, my definition for cardinality requires no extension for infinite sets.

Do you mean that your definition for cardinality is only for finite sets?

 

[jsfisher]

Do you mean that your definition for cardinality is only for finite sets?

Had I meant that, then that would have been what I wrote. It wasn't, so I didn't.

It works for all sets definable under ZF, ZFC, or any other set theory capable of including injective mappings.

 

[Little 10 Toes]

DoronShadmi, Please provide your definition of cardinality. If you don't then we will assume you are accepting marketer's and/or the Wikipedia definition.

 

[Little 10 Toes]

^^^^^Stupid tablet. I mean jsfisher's definition.

 

[doronshadmi]

It works for all sets definable under ZF, ZFC, or any other set theory capable of including injective mappings.

Pleas look at 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.)​

"|A| <= |B|" or "there exists an injection from A to B" are actually tautologies (which means that being finite or non-finite cardinality is determined according to the given set of axioms "under ZF, ZFC, or any other set theory capable of including injective mappings from A to B iff |A| <= |B|.

In that case you need extra mathematical expressions (the axioms of a given set theory) in order to determine whether a given set has finite or non-finite cardinality.

My definition of Cardinality (as given in http://www.internationalskeptics.com/forums/showpost.php?p=13129316&postcount=3507) is a tautology that is independent of any particular set theory, which means that no extra expressions (the axioms) of any given set theory are needed in order to determine whether a given set has finite or non-finite cardinality.

So your tautology needs extra expressions, where mine does not need them.

 

[doronshadmi]

Logical conjunction, for example, does not accept functions for its operands.

Here it is again, following your definition Cardinality:

By injection from domain A to co-domain B, |A| <= |B| (where <= is (< OR =)).

Let X be injection iff X is ((X-only) OR not(X-only))

In first order logic ((not for all X(X=X) OR (for all X(X=X)) is a tautology.

Here is your analogy:

"Not all closed curves are circles, but all circles are closed curves."

By your analogy "closed curves" are equivalent to ((X-only) OR not(X-only)) and "circles" are equivalent not(X-only).

By first order logic "closed curves" are equivalent to "(not for all X(X=X))" and "circles" are equivalent to "(for all X(X=X))"

So according to your analogy:

((X-only) OR not(X-only)) = (not for all X(X=X)), which is False.

OR

not(X-only) = (for all X(X=X)), which is False.

Since the truth value of False OR False is False I don't see how your definition of Cardinality has a logical True basis.

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

Please show my mistake.

Thank you.

 

[jsfisher]

"|A| <= |B|" or "there exists an injection from A to B" are actually tautologies

Apparently you do not know what a tautology is. Given some set A and some set B, |A| <= |B| is a proposition that may or may not be true. That makes it not a tautology. There exists an injection from A to B is a proposition that may or may not be true. That makes it not a tautology.

...which means that being finite or non-finite cardinality is determined according to the given set of axioms "under ZF, ZFC, or any other set theory capable of including injective mappings from A to B iff |A| <= |B|.

Nowhere in the axioms of ZFC, for example, is the meaning of finite cardinality nor infinite cardinality provided. That is something that needs to be added.

My definition of Cardinality...

You haven't provided your definition, yet. You suggested what 'strict cardinality' might mean, but that was not a definition of cardinality.

 

[jsfisher]

Here it is again, following your definition Cardinality:

By injection from domain A to co-domain B, |A| <= |B|

I think you meant |A| <= |B| if and only if there is an injection from A to B.

No need to add extraneous words nor rearrange things to alter the meaning. Mine is a definition of cardinality that does so by defining a relationship (symbolized by <=) between the cardinalities of two sets.

(where <= is (< OR =)).

"< OR =" is not valid anything in predicate calculus. It has no meaning other than your attempt to pretend English language ambiguity can be freely imported into mathematical expression.

Let X be injection iff X is ((X-only) OR not(X-only))

That is complete gibberish.


Be that as it may, how about you spend less time trying to disprove a definition (a fool's errand) and more time providing your definition for cardinality.