Deeper than primes

[doronshadmi]

So? You didn't say you were using the set of natural numbers (usually written as N) and you didn't state you were using cardinal numbers either. Anyways, |F| > any member of F={..., -3, -2, -1} still holds true.

No, it does not hold water (see http://www.internationalskeptics.com/forums/showpost.php?p=4950557&postcount=5499 ).

 

[jsfisher]

Nobody said that there were. Why do you bring up irrelevant points? Are you trying to say the |F| isn't greater than all members of the set F?

http://en.wikipedia.org/wiki/Aleph_number

Aleph-null is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities.

What a truly marvelous talent you have for juxtaposing two statements that aren't related.

So we need two things here which are: 1) Natural Numbers 2) The axiom of choice

Need it for what? By the way, saying that something "is by definition..." isn't the same as saying something "is defined as".

(if there is no natural choice (an example of natural choice is: the smallest member of every nonempty set of natural numbers)).

"if there is no natural choice..." -- yeah, then what? You didn't finish this statement. And for that matter, what does a "natural choice" have to do with the Axiom of Choice?

So by (1) and (2) we have a collection of different finite sizes, where each size is exactly a finite cardinal.

Are you equating the set of natural numbers to a set of sizes? That is totally unnecessary and irrelevant.

However, that isn't relevant. Nothing in your statement, "|N| > any member of N = {1, 2, 3,...}" depends on N being the set of finite cardinals.

Only by using the existence of a collection of infinitely many finite cardinals, we can say that the smallest transfinite cardinal is exactly the cardinal that is greater than any finite cardinal

Nonsense, if for no other reason than all the transfinites are greater than any finite cardinal. The smallest transfinite "isn't exactly" the cardinal that is greater than any finite cardinal; it is not unique in that respect.

You have taken what is fundamentally a simple idea, turned it inside out, then twisted it to match some warped view you have. Stop that.

...or in other words |N| > any member of N, where any member of N does not exist if it is not at least a finite cardinal (ordinals also do not exist without cardinality, in the case of natural numbers, so only cardinality is important here)).

They have existed and will continue to exist without this being a prior condition.

Since |N| > any member of N is the basic rule that enables the existence of |N| as a transfinite cardinal in the first place

It isn't.

it can be generalized to |X| > any member of X iff any member of X is a cardinal (finite or not) and |X| is a transfinite cardinal.

As already demonstrated, this is a false statement. The set of reals, for example, satisfies the left-hand side of your IF-AND-ONLY-IF proposition but not the right.

This generalization does not hold if |X| is a finite cardinal for example: |4| > any member of S={1,2,3,4} is false.

Also F case is not relevant to "|X| > any member of X" definition since F members are not cardinals.

There is no such requirement in your proposition that F consist only of cardinals. You are seeing things that are not there.

There is noting "in general" about it. That would be the set of the finite cardinals. Note, too, that while you might consider this weakly to be a definition of a set we'll call N, it is not a definition of the set of finite cardinals.

"|X| > any member of X" is relevant as general definition for transfinite cardinals only if the members of the collection are themselves cardinals and |X| is a transfinite cardinal.

In the real world of mathematics, this isn't the case.

In general, there is no general extension form the finite to the non-finite, and this is exactly why Cantor's reasoning fails (he forces the finite in order to define the non-finite. We do not need more that Dedekind-infinite ( http://en.wikipedia.org/wiki/Dedekind-infinite_set ) in order to show that there cannot be such an extension).

Which part, exactly, did Cantor get wrong?

It's great how you just make this crap up, doron, without much thought about what you are actually saying. Your statement wasn't even in the correct grammatical form to be a definition. Be that as it may, what in your "definition" constrains |N| to be exactly Aleph-null? Not much of a definition, now is it, if it doesn't actually make something specific.

I show that Cantor's reasoning does not hold water, because it is based on forcing the finite on the non-finite.

No, you showed again how little you understand English, Mathematics terminology, logic, and reasoning.

Did you perhaps want to say something like "|N| is the smallest transfinite cardinal number, where N is the set of positive integers"? You didn't. You didn't even come close.

No, I want to say that N collection simply does not exist if each member of it is not at least a finite cardinal > 0.

So, the set N = {0, 1, 2, 3} does not exist. That's an interesting new set theory you have there.

 

[doronshadmi]

What a truly marvelous talent you have for juxtaposing two statements that aren't related.

The maneuver is exactly the result of your serial step-by-step fragmented reasoning style. You reply before you read all of what is written.

Need it for what? By the way, saying that something "is by definition..." isn't the same as saying something "is defined as".

Did any one here started here to talk about maneuvers?

Are you equating the set of natural numbers to a set of sizes? That is totally unnecessary and irrelevant.

No just sizes, N members are exactly the finite cardinals, where each member > 0 (furthermore, in the case of natural numbers, ordinals depends on the existence of cardinals and not vise versa, so finite cardinality is the fundamental thing that enables N existence, in the first place).

Nonsense, if for no other reason than all the transfinites are greater than any finite cardinal. The smallest transfinite "isn't exactly" the cardinal that is greater than any finite cardinal; it is not unique in that respect.

Nonsense, the smallest transfinite cardinal is exactly the cardinal that is greater only than any finite cardinal. The rest of transfinite cardinals are greater also than other transfinite cardinals, which is a property that the smallest transfinite cardinal does not have.

You have taken what is fundamentally a simple idea, turned it inside out, then twisted it to match some warped view you have. Stop that.

Forcing the finite on the non-finite, this is fundamentally a twisted idea, and all your reasoning is based on this false forcing.

They have existed and will continue to exist without this being a prior condition.

This is a fundamental condition. Just take finite cardinality out of existence and see what happens to N existence.

It isn't.

Yes it is.

As already demonstrated, this is a false statement. The set of reals, for example, satisfies the left-hand side of your IF-AND-ONLY-IF proposition but not the right.

Nothing was demonstrated, |R| > than the cardinality of any member of the power set of N. Remember |X| > any member of X, has a meaning in the first place only if the cardinality of each X member is compared (and not the structure of the member) to |X|, which is a transfinite cardinal.

There is no such requirement in your proposition that F consist only of cardinals. You are seeing things that are not there.

No you force things on my definition that are not there. Again |X| > any member of X iff the cardinality of any member of X is compared with |X| and |X| is a transfinite cardinal.

In the real world of mathematics, this isn't the case.

It is a fantasy world because you force the finite on the non-finite.

Which part, exactly, did Cantor get wrong?

His notion to close the non-finite in some fixed box, as can be done to finite cases.

No, you showed again how little you understand English, Mathematics terminology, logic, and reasoning.

You have shown again how your reasoning is based on illusions, which is a direct result of: 1) Getting everything by fragmented step-by-step reasoning 2) Forcing the finite on the non-finite.

So, the set N = {0, 1, 2, 3} does not exist. That's an interesting new set theory you have there.

No, {0,1,2,3} is not set N.

 

[doronshadmi]

Let us penetrate to the heart of this discussion:

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

Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel.

Cantor an his friends, did not understand the non-finite because they did not understand that no collection of parts can be the whole.

In order to get it a Direct Perception of the researched is needed, which is more accurate than any intuition or logical reasoning, simply because it is the fundamental source of both of them, and the non-finite is not understood without Direct Perception.

Let us use, for example the, Hilbert's paradox of the Grand Hotel.

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

This provides an important and non-intuitive result; the situations "every room is occupied" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.

"every room is occupied" is some locality, and no infinitely many localities are Non-locality (The existence of the path that enables some room, has nothing to do with the existence of the gusts on it). Therefore there is always a room for more accommodations, exactly because Hilbert's Grand Hotel is the exact result of Non-locality\Locality Linkage.

 

[The Man]

(You have a typo mistake, it has to be ">" and not "<")

You have missed the part that:

|K| > |N| exactly by 1

|L| > |N| excetly by 2

etc ...


We have here different transfinite cardinals, yet they are exist by the same general definition, which enables transfinite cardinals (no matter what their size are) in the first place.

Nope the mistakes are entirely yours, again.

In order for |K| and |L| not to meet your given definition of the “the smallest transfinite cardinal” which is simply that it is " > any member of N={1,2,3,…}” then |K| and |L| must be ≤ “any member of N={1,2,3,…}”.

This:

|X| > any member of X is a general definition of any transfinite cardinal, whether X members are transfinite cardinals or not. <snip>

So since |X| is > any member of X = {0, 1, 2, 3} that makes |X| a “transfinite cardinal” by your above “general definition”. Your ‘general definitions’ are just general nonsense.

 

[The Man]

<snip>

Since |N| > any member of N is the basic rule that enables the existence of |N| as a transfinite cardinal in the first place, it can be generalized to |X| > any member of X iff any member of X is a cardinal (finite or not) and |X| is a transfinite cardinal.
<snip>

So now your purported "general definition" of a “transfinite cardinal” now simply requires |X| to be a “transfinite cardinal” in order for you to ‘generally define’ it as a “transfinite cardinal”. You continue to astound only yourself.

 

[doronshadmi]

Nope the mistakes are entirely yours, again.

In order for |K| and |L| not to meet your given definition of the “the smallest transfinite cardinal” which is simply that it is " > any member of N={1,2,3,…}” then |K| and |L| must be ≤ “any member of N={1,2,3,…}”.



So since |X| is > any member of X = {0, 1, 2, 3} that makes |X| a “transfinite cardinal” by your above “general definition”. Your ‘general definitions’ are just general nonsense.

The Man you are running after your own problems, because:

1) My definition is:

|X| > any member of X iff the cardinality of any member of X is compared with |X|, and |X| is a transfinite cardinal.

and you ignore my definition.

2) |K| and |L| can't be the smallest transfinite cardinal exactly because K and L have members that are transfinite cardinals (which is something that is false in the case of |N| (N does not have any member, which is a transfinite cardinal, and this is exactly the reason of why |N| is the smallest transfinite cardinal)).

 

[doronshadmi]

So now your purported "general definition" of a “transfinite cardinal” now simply requires |X| to be a “transfinite cardinal” in order for you to ‘generally define’ it as a “transfinite cardinal”. You continue to astound only yourself.

It is a general definition that holds only in the case of transfinite cardinals, so?

There is no circularity here exactly as {} is one of the sets that are not members of {}, in the definition of the Empty set.

Furthermore, a general definition of A is anything that is relevant only to A, and A in this case is a transfinite cardinal, so?

 

[The Man]

The Man you are running after your own problems, because:

1) My definition is:

|X| > any member of X iff the cardinality of any member of X is compared with |X|, and |X| is a transfinite cardinal.

Definition of what? It can’t be a ‘definition’ of a “transfinite cardinal” since you are requiring “|X|” to be a “transfinite cardinal” in order for you to define it as a “transfinite cardinal”. Your so called “general definition” simply amounts to you stating and requiring that “|X| is a transfinite cardinal”.

and you ignore my definition.

2) |K| and |L| can't be the smallest transfinite cardinal exactly because K and L have members that are transfinite cardinals (which is something that is false in the case of |N| (N does not have any member, which is a transfinite cardinal, and this is exactly the reason of why |N| is the smallest transfinite cardinal)).

Definition of what? Your given definition of “the smallest transfinite cardinal” was simply that it is “> any member of N={1, 2, ,3}”, “|K| and |L|” meet that “definition”. The fact that your ‘general definitions’ are just general nonsense is no ones problem but your own. Asserting, as you do above, that your “general definition” is insufficient on its own simply demonstrates that you know it is crap yourself, yet still posit it as meaningful anyway.

 

[doronshadmi]

You know what The Man, let us do it simpler:

X is a non-finite collection.

General definition:

If |X| > the cardinality of any member of X, then |X| is a transfinite cardinal.

The smallest transfinite cardinal:

If |X| > the cardinality of any member of X and the cardinality of any member of X is finite, then |X| is the smallest transfinite cardinal.

The cardinality of each member is not |{x}| but it is |x|.

|K| (where K={N,1,2,3,...}) in not the smallest transfinite cardinal, because N (and therefore |N|) is one of its compared values.

 

[The Man]

It is a general definition that holds only in the case of transfinite cardinals, so?

There is no circularity here exactly as {} is one of the sets that are not members of {}, in the definition of the Empty set.

Furthermore, a general definition of A is anything that is relevant only to A, and A in this case is a transfinite cardinal, so?

Again the fact that you find yourself claiming your “general definition” of “transfinite cardinals” “holds only in the case of transfinite cardinals” simply demonstrates that your “general definition” is meaningless.

 

[The Man]

You know what The Man, let us do it simpler:

X is a non-finite collection.

General definition:

If |X| > the cardinality of any member of X, then |X| is a transfinite cardinal.

The smallest transfinite cardinal:

If |X| > the cardinality of any member of X and the cardinality of any member of X is finite, then |X| is the smallest transfinite cardinal.

Nope if the members of "X" are not sets then they do not have cardinality.

You know Doron, let's do it even simpler and just use the actual definitions for cardinality.

 

[jsfisher]

Nonsense, if for no other reason than all the transfinites are greater than any finite cardinal. The smallest transfinite "isn't exactly" the cardinal that is greater than any finite cardinal; it is not unique in that respect.

Nonsense, the smallest transfinite cardinal is exactly the cardinal that is greater only than any finite cardinal. The rest of transfinite cardinals are greater also than other transfinite cardinals, which is a property that the smallest transfinite cardinal does not have.

I see you have added a word that wasn't there before. Will you be moving the goal posts again any time soon?

They have existed and will continue to exist without this being a prior condition.

This is a fundamental condition. Just take finite cardinality out of existence and see what happens to N existence.

If by N, you mean the set of natural numbers, then the set N exists as a direct consequence of the Axiom of Infinity, not cardinality. Meanwhile, the cardinality concept can be developed independent of the natural numbers.

As already demonstrated, this is a false statement. The set of reals, for example, satisfies the left-hand side of your IF-AND-ONLY-IF proposition but not the right.

Nothing was demonstrated, |R| > than the cardinality of any member of the power set of N.

Again, you see things that aren't there. And again you change things from what are stated. I claim |R| > any member of R, where R is the set of real numbers. Are you claiming that is a false statement? Is there some real number that is greater than the cardinality of R?

Remember |X| > any member of X, has a meaning in the first place only if the cardinality of each X member is compared (and not the structure of the member) to |X|, which is a transfinite cardinal.

No, the mathematical statement |X| > any member of X is simply a proposition. It is either true or false depending on the contents of set X. All that other stuff about cardinality is just junk you invented out of thin air.

There is no such requirement in your proposition that F consist only of cardinals. You are seeing things that are not there.

No you force things on my definition that are not there. Again |X| > any member of X iff the cardinality of any member of X is compared with |X| and |X| is a transfinite cardinal.

No, I insist you not pretend things are in your definition that are not. The left-hand side of your IFF proposition is the simply proposition |X| > any mbmer of X. For any given set, the proposition is either true or false. There are no constraints placed on the contents nor size of the set.

Which part, exactly, did Cantor get wrong?

His notion to close the non-finite in some fixed box, as can be done to finite cases.

Can you be a bit more specific?

So, the set N = {0, 1, 2, 3} does not exist. That's an interesting new set theory you have there.

No, {0,1,2,3} is not set N.

What, you think there is only one possible use for the letter N that's allowed? N = {0, 1, 2, 3} is a perfectly acceptable labeling for a set.

 

[doronshadmi]

Nope if the members of "X" are not sets then they do not have cardinality.

You know Doron, let's do it even simpler and just use the actual definitions for cardinality.

Your are wrong:

1={{}}

2={{},{{}}}

3={{},{{}},{{},{{}}}}

...

N={1,2,3,...}

K={N,1,2,3,...}

X is a non-finite collection.

General definition:

If |X| > the cardinality of any member of X, then |X| is a transfinite cardinal.

The smallest transfinite cardinal:

If |X| > the cardinality of any member of X and the cardinality of any member of X is finite, then |X| is the smallest transfinite cardinal.

The cardinality of each member is not |{x}| but it is |x| where x is a set (exactly as shown above).

|K| (where K={N,1,2,3,...}) is not the smallest transfinite cardinal, because N (and therefore |N|=|{1,2,3,...}|) is one of its compared values.

There is no problem to reduce any member of R set to collection of sets, etc ... as shown above.

 

[jsfisher]

Let us penetrate to the heart of this discussion:

Cantor an his friends, did not understand the non-finite because they did not understand that no collection of parts can be the whole.

And yet there are just as many even integers as there are integers. Hmm. Imagine that.

 

[doronshadmi]

And yet there are just as many even integers as there are integers. Hmm. Imagine that.

In general, there is no general extension form the finite to the non-finite, and this is exactly why Cantor's reasoning fails (he forces the finite in order to define the non-finite. We do not need more than Dedekind-infinite ( http://en.wikipedia.org/wiki/Dedekind-infinite_set ) in order to show that there cannot be such an extension).

N and E (where E is the collection of even numbers) are nothing but non-finite comparison between localities , and no collection of localities is the whole (which is non-local).

 

[The Man]

Your are wrong:

1={{}}

2={{},{{}}}

3={{},{{}},{{},{{}}}}

...

If you are trying to talk about the cardinality of what looks like you’re trying to represent as sets, you are still wrong. The cardinality of a set is not itself a set.

N={1,2,3,...}

K={N,1,2,3,...}

X is a non-finite collection.

General definition:

If |X| > the cardinality of any member of X, then |X| is a transfinite cardinal.

The smallest transfinite cardinal:

If |X| > the cardinality of any member of X and the cardinality of any member of X is finite, then |X| is the smallest transfinite cardinal.

The cardinality of each member is not |{x}| but it is |x| where x is a set (exactly as shown above).

|K| (where K={N,1,2,3,...}) is not the smallest transfinite cardinal, because N (and therefore |N|) is one of its compared values.

Again if the members of X are not sets then they do not have cardinality. You have previously said that the members of your set X, N and K sets are cardinal numbers which means that your required members are specifically not sets and thus do not themselves have cardinality.

 

[jsfisher]

A couple of technical points:

N={1,2,3,...}

The more common convention is for the set of natural numbers to include zero.

X is a non-finite collection.

If |X| > the cardinality of any member of X, then |X| is a transfinite cardinal.

I see we have added more words trying to correct previous bogus statements, but this is still not a definition. It's a tautology. In fact, that whole middle part is unnecessary. It can be reduced to:

If X is an infinite set, then |X| is a transfinite cardinal​

The smallest transfinite cardinal:

If |X| > the cardinality of any member of X and the cardinality of any member of X is finite, then |X| is the smallest transfinite cardinal.

This one is just plain wrong.

 

[jsfisher]

In general, there is no general extension form the finite to the non-finite, and this is exactly why Cantor's reasoning fails (he forces the finite in order to define the non-finite.

And where did this forcing take place? Please be specific.

N and E (where E is the collection of even numbers) are nothing but non-finite comparison between localities , and no collection of localities is the whole (which is non-local).

So you are saying there are fewer even numbers that integers. Interesting point of view you have, there. Totally unsupported by the facts, but interesting nonetheless.

 

[jsfisher]

The smallest transfinite cardinal:

If |X| > the cardinality of any member of X and the cardinality of any member of X is finite, then |X| is the smallest transfinite cardinal.

This one is just plain wrong.

In anticipation of the now-sleeping Doron's vacuous protestations...

Let I be the minimal set guaranteed by the Axiom of Infinity.
Let P be the power set of I.
Then, let X be the set { {a} : a ∈ P }.