Deeper than primes

[doronshadmi]

I agree with jsfisher on this one. You have only done a statement. Here, I'll do one too!

|F| > any member of F={..., -3, -2, -1}

Same, same only different.

There are no negative cardinals.

In general N={1,2,3,...} is also the collection of finite cardinals, and by using this fact |N| > any member of N={1,2,3,...} is exactly the definition of the smallest transfinite cardinal.

 

[doronshadmi]

Since |R| meets the given requirements of “> any member of N={1, 2, 3, …}” then |R| must meet the “definition of the smallest transfinite cardinal” and |R| = |N| by Doron’s given requirements.

No.

I show that |K| > any member of K={|N|,1,2,3,...}, and |R| > |K| > |N| so your argument does not hold.

 

[The Man]

No.

I show that |K| > any member of K={|N|,1,2,3,...}, and |R| > |K| > |N| so your argument does not hold.

"Definition" is yours Doron, so your "definition" does no hold, even just for you.

 

[doronshadmi]

"Definition" is yours Doron, so your "definition" does no hold, even just for you.

No, it is a generalization of the definition of transfinite cardinals, starting by |N|:

|N| > any member of N={1,2,3,…} (where N is a collection of finite cardinals and this is exactly the reason of why |N| is the smallest transfinite cardinal).

|K| > any member of K={|N|,1,2,3,…}, therefore |K| > |N| by 1.

|L| > any member of L={|K|,|N|,1,2,3,…}, therefore |L| > |N| by 2, and |L| > |K| by 1.

Etc …

 

[doronshadmi]

As do I.

Let's try a different tact

My cat has four legs.

A true statement.

Does having four legs define my cat?

At least in part, but my neighbor’s dog also has four legs and certainly is not my cat. Having four legs is just a property of my cat as well as my neighbor’s dog.

Does my cat define the property of having four legs?

Since cardinality is an abstraction of some structure into amounts (finite or not), then yes, by reducing the researched structure to amounts, there is a generalization of cats and dogs to amounts.

 

[The Man]

No, it is a generalization of the definition of transfinite cardinals, starting by |N|:

|N| > any member of N={1,2,3,…} (where N is a collection of finite cardinals and this is exactly the reason of why |N| is the smallest transfinite cardinal).

|K| > any member of K={|N|,1,2,3,…}, therefore |K| > |N| by 1.

|L| > any member of L={|K|,|N|,1,2,3,…}, therefore |L| > |N| by 2, and |L| > |K| by 1.

Etc …

Oh, so it is a “generalization of the definition of transfinite cardinals”. I should have guessed. I also see that you are employing your standard interpretation of “generalization” meaning that your “definition” is self inconsistent and even what you are not ‘defining’ as “the smallest transfinite cardinal” meets your “definition” of “the smallest transfinite cardinal“. As ‘defined’ above, |K| > any member of N={1,2,3,…} and |L| > any member of N={1,2,3,…} thus both meet your “generalization of the definition” of “the smallest transfinite cardinal”. Claiming “generalization” does not make your “definition” any more self consistent then it was before.

 

[doronshadmi]

Oh, so it is a “generalization of the definition of transfinite cardinals”. I should have guessed. I also see that you are employing your standard interpretation of “generalization” meaning that your “definition” is self inconsistent and even what you are not ‘defining’ as “the smallest transfinite cardinal” meets your “definition” of “the smallest transfinite cardinal“. As ‘defined’ above, |K| > any member of N={1,2,3,…} and |L| > any member of N={1,2,3,…} thus both meet your “generalization of the definition” of “the smallest transfinite cardinal”. Claiming “generalization” does not make your “definition” any more self consistent then it was before.

No, |K|, |L| etc ... do not meet my “definition” of “the smallest transfinite cardinal“ because K or L etc ... have also transfinite cardinals as their members, which is something that N does not have.

But all of them are based on the same generalization that enables to define transfinite cardinals, whether they are the smallest transfinite cardinal (which is true only in |N| case) or not.

 

[The Man]

No, |K|, |L| etc ... do not meet my “definition” of “the smallest transfinite cardinal“ because K or L etc ... have also transfinite cardinals as their members, which is something that N does not have.

But all of them are based on the same generalization that enables to define transfinite cardinals, whether they are the smallest transfinite cardinal (which is true only in |N| case) or not.

So the inclusion of "transfinite cardinals as their members" makes "|K|, |L| etc ..." < “any member of N={1,2,3,…}”?

 

[doronshadmi]

So the inclusion of "transfinite cardinals as their members" makes "|K|, |L| etc ..." < “any member of N={1,2,3,…}”?

(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.

 

[zooterkin]

No, it is a generalization of the definition of transfinite cardinals, starting by |N|:

|N| > any member of N={1,2,3,…} (where N is a collection of finite cardinals and this is exactly the reason of why |N| is the smallest transfinite cardinal).

|K| > any member of K={|N|,1,2,3,…}, therefore |K| > |N| by 1.

Why 'therefore'? What forces this?

 

[doronshadmi]

Why 'therefore'? What forces this?

This:

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

By using this general definition (that enables transfinite cardinals, in the first place) we are enable to distinguish between collections that do not have transfinite cardinals as their members, and collections that have transfinite cardinals as their members.

Again, it has no be stressed that without "|X| > any member of X" definition, transfinite cardinals (no matter what their size are) do not exist.

 

[zooterkin]

This:

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

By using this general definition (that enables transfinite cardinals, in the first place) we are enable to distinguish between collections that do not have transfinite cardinals as their members, and collections that have transfinite cardinals as their members.

Again, it has no be stressed that without "|X| > any member of X" definition, transfinite cardinals (no matter what their size are) do not exist.

You miss the point. Why "therefore |K| > |N| by 1." Where did the 'by 1' come from?

 

[doronshadmi]

"|X| > any member of X" means that any X member is taken as some cardinal (for example ( {{}} and the power of {} are actually cardinal 1, or {{},{{}}, {{},{{}}}} and 3 are actually cardinal 3, etc …).

In other words, any member is reduced to its cardinality, which is something that is different than the notion of collections and power collections, as used by Standard Math.

 

[doronshadmi]

You miss the point. Why "therefore |K| > |N| by 1." Where did the 'by 1' come from?

No, you miss the point of the definition that enables the existence of transfinite cardinals, which is exactly "|X| > any member of X".

Without it, you have nothing to talk about including your question.


So first you have to ask yourself:

What enables the existence of transfinite cardinals, in the first place?


The answer is exactly: "|X| > any member of X"

By using this definition, we wish to define the difference between transfinite cardinals, and it must follow the fact that the smallest transfinite cardinal is definable only if X members are reduced to sizes (their internal structures are ignored and generalized to some size).

This is exactly what we are doing by define the smallest transfinite cardinal, which its size is bigger than any finite size.

It is notated as |N| > any member of N={|1|,|2|,|3|,…} where N is a collection of finite sizes (we ignore the infinitely many structures that each size represent if we really wish to be consistent about the whole idea of researching Size, in the first place).

By using this consistent reasoning we research the difference of the sizes between cardinals (whether they are finite or not).

By using this consistent reasoning (that is not based on the mapping technique) it is obvious that |K| > |N| by 1, exactly because K has one more transfinite member than N has.

 

[jsfisher]

There are no negative cardinals.

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?

In general N={1,2,3,...} is also the collection of finite cardinals

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.

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.

...and by using this fact |N| > any member of N={1,2,3,...} is exactly the definition of the smallest transfinite cardinal.

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.

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.

 

[jsfisher]

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

Got any legitimate reference to support this bogus assertion of yours? Of course not.

 

[jsfisher]

So first you have to ask yourself:

What enables the existence of transfinite cardinals, in the first place?

The correct answer would be Georg Cantor. Luckly, Cantor was completely unaware of Doron's claimed "definition", and thus able to develop the mathematics of the transfinite without handicap.

 

[Little 10 Toes]

There are no negative cardinals.

In general N={1,2,3,...} is also the collection of finite cardinals, and by using this fact |N| > any member of N={1,2,3,...} is exactly the definition of the smallest transfinite cardinal.

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.

 

[doronshadmi]

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.

So we need two things here which are: 1) Natural Numbers 2) The axiom of choice (if there is no natural choice (an example of natural choice is: the smallest member of every nonempty set of natural numbers)).

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

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, 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)).

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.

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 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 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).

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.

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.

 

[doronshadmi]

The correct answer would be Georg Cantor. Luckly, Cantor was completely unaware of Doron's claimed "definition", and thus able to develop the mathematics of the transfinite without handicap.

Poor Cantor created the bogus CH exactly because he forced the finite on the non-finite, and as a result missed the whole point about the non-finite.


You are a proud scholar to this bogus reasoning, no more no less.