Deeper than primes

[doronshadmi]

By the construction of set X, we know |X| = |P|. Since |P| > |I|, we know |X| > |I|. Since sets I and X are both infinite sets, and |X| > |I|, |X| cannot be the smallest transfinite cardinal.

Please show some example of I set.

For example, if I={a,b,c,...} and X={{a},{b},{c},...}

Then:

1) There is no problem to show that:

 X       I

{a} <--> a
{b} <--> b
{c} <--> c
...

2) X set is not the powerset of I.

 

[jsfisher]

Please show some example of I set.

Your favorite set, N, is a fine example.

 

[doronshadmi]

Your favorite set, N, is a fine example.

(No, please read it again)

Please show some example of I set.

For example, if I={a,b,c,...} and X={{a},{b},{c},...}

Then:

1) There is no problem to show that:

 X       I

{a} <--> a
{b} <--> b
{c} <--> c
...

2) X set is not the powerset of I.

 

[jsfisher]

(No, please read it again)

Please show some example of I set.

For example, if I={a,b,c,...} and X={{a},{b},{c},...}

The set X is not defined that way. X = {{a} : a ∈ P } where P is the power set of I.

...
2) X set is not the powerset of I.

...nor is it supposed to be.

 

[doronshadmi]

The set X is not defined that way. X = {{a} : a ∈ P } where P is the power set of I.



...nor is it supposed to be.

I={a,b,c,...}

X={{a},{b},{c},...} by your construction.

and it is trivial to show that:

 X       I

{a} <--> a
{b} <--> b
{c} <--> c
...

So |X| = |I| and |X| is the smallest trasfinite cardinal.

So, all the conditions you laid out are satisfied, but |X| is not the smallest transfinite cardinal number.

Wrong.

 

[jsfisher]

I={a,b,c,...}

X={{a},{b},{c},...} by your construction.

No!

If I = {a, b, c, ...} then P = {{}, {a}, {a,b}, {b}, {a,b,c}, ...} and then X = { {{}}, {{a}}, {{a,b}}, {{b}}, {{a,b,c}}, ...}

 

[doronshadmi]

No!

If I = {a, b, c, ...} then P = {{}, {a}, {a,b}, {b}, {a,b,c}, ...} and then X = { {{}}, {{a}}, {{a,b}}, {{b}}, {{a,b,c}}, ...}

Very nice jsfisher

|{a,b,c,...}| is a transfinite cardinal.

|{{a,b,c,...}}| is a finite cardinal.

So the whole idea of Cardinality is based on the first level of some set.

In that case my notion about Cardinality does not hold, but we can ask:

1) Is it because it is too complex for the minds of the mathematicians?

2) Is there some "deep" reason of why Cardinality is the measurement of the size of the first-level of some set (is it because it is too complex for the minds of the mathematicians)?

If the answer is NO to (1) AND NO to (2), then there can be another reasoning about sets that does not stop at the first level. In that case non-finite sets cannot be hidden behind the first level.

 

[jsfisher]

Very nice jsfisher

|{a,b,c,...}| is a transfinite cardinal.

|{{a,b,c,...}}| is a finite cardinal.

Dah! That was the whole point.

So the whole idea of Cardinality is based on the first level of some set.

Dah! If you actually looked up the definition of cardinality instead of trying to invent your own, you might have realized this fact sooner.

In that case my notion about Cardinality does not hold

Yeah, I think several people have been trying to point that out to you.

...
but we can ask:

1) Is it because it is too complex for the minds of the mathematicians?

Hardly.

2) Is there some "deep" reason of why Cardinality is the measurement of the size of the first-level of some set

Yes.

(is it because it is too complex for the minds of the mathematicians)?

Not bloody likely.

If the answer is NO to (1) AND NO to (2), then there can be another reasoning about sets that does not stop at the first level. In that case non-finite sets cannot be hidden behind the first level.

You missed the obvious reason.

 

[doronshadmi]

So by the standard paradigm of the concept of Set there are two basic assumptions:

1) The size of a set is measured only at its first level of complexity.

2) But in order to distinguish between sets no level of complexity is ignored, for example:

{{a,b,c,…}} is not {a,b,c,…} exactly because no level of complexity is ignored, and this difference is translated to different cardinalities, by the standard paradigm.

By the way:

|{{a,b,c,…}}|=1 means that {a,b,c,…} is taken as a whole where the parts are ignored.

|{a,b,c,…}|=|N| means that {a,b,c,…} is taken as parts where the whole is ignored.

So, what will happen to Set Theory if both Whole AND Parts are not ignored?

In that case the "deep" reasoning of Complexity gets on stage and we immediately understand (by using Direct Perception) that 1 of 1= |{{a,b,c,…}}| is actually the measurement of the whole and |N| of |N|=|{a,b,c,…}| is actually the measurement of the parts of the same Complexity.

In that case 1 and |N| are actually measurements of two different qualities of Complexity (which is something that the current paradigm misses, because it understands them as two different quantities).

 

[jsfisher]

So by the standard paradigm of the concept of Set there are two basic assumptions:

1) The size of a set is measured only at its first level of complexity.

No. The basic assumptions about sets are contained in the axioms of set theory. Cardinality, on the other hand, is a separate concept introduced specifically to compare sizes of sets (in terms of quantity of members).

2) But in order to distinguish between sets no level of complexity is ignored

No, cardinality does not play a part in distinguishing between sets. All that is required is to find a difference in the membership of the respective sets.

for example:

{{a,b,c,…}} is not {a,b,c,…} exactly because no level of complexity is ignored, and this difference is translated to different cardinalities, by the standard paradigm.

No, for the reason already given.

By the way:

|{{a,b,c,…}}|=1 means that {a,b,c,…} is taken as a whole where the parts are ignored.

No, cardinality is a measure of size of a set, nothing more and nothing less. The "parts" aren't being ignored, but it is only there existence that is relevant to the measure.

|{a,b,c,…}|=|N| means that {a,b,c,…} is taken as parts where the whole is ignored.

No. In both the {a,b,c,...} case and the {{a,b,c,...}} case, we are considering exactly the same concept, namely the size of the set in terms of the number of elements the set contains.

So, what will happen to Set Theory if both Whole AND Parts are not ignored?

I am not entirely sure, but I am betting you will end up with some fairly worthless concepts.

In that case the "deep" reasoning of Complexity gets on stage and we immediately understand (by using Direct Perception) that 1 of 1= |{{a,b,c,…}}| is actually the measurement of the whole and |N| of |N|=|{a,b,c,…}| is actually the measurement of the parts of the same Complexity.

Only if you want to misinterpret things. Cardinality does not measure complexity.

In that case 1 and |N| are actually measurements of two different qualities of Complexity (which is something that the current paradigm misses, because it understands them as two different quantities).

No. The "current paradigm" doesn't miss anything of the sort. Cardinality is not a measure of complexity. It is your misunderstanding for interpreting it as such, then for faulting it for not meeting your expectations.

 

[doronshadmi]

No, cardinality does not play a part in distinguishing between sets. All that is required is to find a difference in the membership of the respective sets.

No, cardinality is a measure of size of a set, nothing more and nothing less. The "parts" aren't being ignored, but it is only there existence that is relevant to the measure.

Cardinality is impossible without an existing thing that enables to measure the existence of things, such that the existence of that thing is stronger than any measurement result.

For example: The existence of {} is stronger than the measurement result, which is 0=|{}| and this fact is not changed even if infinitely many things are measured, for example |R| = |{ {{}}, {{a}}, {{a,b}}, {{b}}, {{a,b,c}}, ...}|

Without this stronger existence there is no way to find the difference between sets according to the membership of the respective sets, or in other words Cardinality is impossible.

The mathematical science is possible in the first place, exactly because it uses this stronger existence, and it does the right thing by not measure the stronger existence as one of the measured things.

But (to use X) AND (to ignore X) is a contradiction, or in other words, the mathematical science is based on contradiction as long as X is used AND ignored.

OM solves this contradiction by using Direct Perception, which enables to define the stronger existence
as Non-locality and the measured things as Localities, that if gathered by Non-locality their existence is less than Non-locality.

By doing that OM enables to deal with Complexity.

Again:

The existence of the Non-local Path does not depend on the existence of infinitely many Localities on it, and it can be demonstrated right at {}, such that the cardinality of {} (=0) is less than the existence of {}.

 

[jsfisher]

Cardinality is impossible without an existing thing that enables to measure the existence of things, such that the existence of that thing is stronger than any measurement result.

You seem to be mistaking consistency with causality.

...snip of stuff using the undefined term, stronger...

Define stronger.

But (to use X) AND (to ignore X) is a contradiction, or in other words, the mathematical science is based on contradiction as long as X is used AND ignored.

And just where do you see that being done? Please be specific.

OM solves this contradiction by using Direct Perception, which enables to define the stronger existence
as Non-locality and the measured things as Localities, that if gathered by Non-locality their existence is less than Non-locality.

Doron, so far, you haven't successfully defined anything, so how can you make this empty claim, here? Moreover, look how poorly your direct perception served you with your bogus definitions for transfinite cardinals.

Every example you try for the wonders of direct perception or of organic mathematics has failed. Got anything that actually works?

By doing that OM enables to deal with Complexity.

How? Please be specific. Or is this something you will be adding soon then claiming it was there right along?

Again:

The existence of the Non-local Path does not depend on the existence of infinitely many Localities on it, and it can be demonstrated right at {}, such that the cardinality of {} (=0) is less than the existence of {}.

Define path. How does this less-than operator of yours work when comparing existences?

 

[zooterkin]

I think I may have nodded off at some point. Do we know where this is going, or is Doron just attempting to reproduce the definitions of set theory by trial and error?

 

[doronshadmi]

Doron, so far, you haven't successfully defined anything, so how can you make this empty claim, here? Moreover, look how poorly your direct perception served you with your bogus definitions for transfinite cardinals.

Jsfisher, as you know I understand Cardinality different than you.

For me Cardinality is the measurement of the existence of things, and this measurement is not limited to any particular level of the measured thing (in your case Cardinality is the measurement of the existing sets that are defined at the first level of sets).

In that case my (3) definition does not hold under your arbitrary limitations.

Let us examine it more precisely.

Define stronger.

Very simple: {} is an existing thing but its Cardinality = 0.

By using Direct Perception we immediately understand that the existence of {} is stronger than the existence of the measured members that are related to it. In this case no member is related to {} (including itself) and yet it exists independently of the luck of members. So there is a difference between being Membership and being Member, where the existence of Membership is stronger than or independent of the existence of members.

I call Membership Non-locality and Member Locality, and also show that no amount of Localities is Non-locality.

This fact (The independent existence of Membership of any Member(s)) is used AND ignored by Standard Math, which is a contradiction that is avoided by OM.

Because of this -used AND ignored Standard Math contradiction- things like {{a,b,c,…}} and {a,b,c,…} are not fully understood, for example:

Cardinality 1=|{{a,b,c,…}}| is based on the existence of {a,b,c,…} by ignoring the understanding of the building-blocks that enable this existence, in the first place.

As a result the difference of Complexity between {{x}} and {x} is not learned as a significant property of the researched subjects, and things are trivialized into member's amounts without the notion of how their existence is possible, in the first place.

If one does not understand the Building-Blocks of X, one does not understand X, and therefore does not understand any full result of X.

Let us demonstrate it by using {{a,b,c,…}}.

The 1 of 1=|{{a,b,c,..}}| is possible in the first place exactly because the measured object is the result of the linkage between different levels of existence, symbolized as "{""}" and "a,b,c,…", where the existence of "{""}" is stronger than the existence of "a,b,c,…" (as clearly demonstrated by {} case).

It does not matter what symbols are used, for example:

"A" of {A} is equivalent to the internal "{""}" of {{a,b,c,…}}.

Cantor, by using exactly the fact that <--> (mapping) existence is stronger than the existence of the mapped objects (<--> exists even if there is no input\output, exactly as {} exists without members) uses this fact in order to show that he enables to define a member of the power set of X that is not mapped with any member of set X, which enables to conclude that |P(X)| > |X|.

This result is based on measuring only the first level of sets (by using AND ignoring what actually enables this measurement, which is not less than Membership(Non-locality)\Members(Localities) Linkage) and any other measurement that is not based on the first level, is arbitrarily ignored.

Since I use Direct Perception, I get sets Cardinality by not ignoring their Complexity as seen from Membership(Non-locality)\Members(Localities) Linkage.

From this "X-ray"-like comprehensive point of view, let us re-examine the following:

1) X is a set and any member of X (if exists) is a set.

2) If X is an infinite set, then |X| is a transfinite cardinal.

3) If |X| is a transfinite cardinal, such that |X| > the cardinality of any member of X and any member of X is a finite set, then |X| is the smallest transfinite cardinal.

It is shown by jsfisher that (3) does not hold if Cardinality is a measurement that is limited only to the first level of sets.

But (3) holds if Cardinality is the measurement of the existence of anything that is the result of Membership(Non-locality)\Members(Localities) Linkage.

In that case the Cardinality of X={{a,b,c,…}} is |N|=|{a,b,c,…}| + 1 and (3) holds.


In other words jsfisher "You win the battle but lose the war" exactly because you are using a reasoning that is based on arbitrary limitations.

 

[doronshadmi]

|{}|=0

|{{}}|= 0+1

|{{{}}}| = 0+1+1

...

|{{a,b,c,…}}| is |N|=|{a,b,c,…}| + 1


...

 

[jsfisher]

Jsfisher, as you know I understand Cardinality different than you.

Yes, I think everyone here realizes you just make stuff up because you don't understand Mathematics nor its terminology.

For me Cardinality is the measurement of the existence of things, and this measurement is not limited to any particular level of the measured thing (in your case Cardinality is the measurement of the existing sets that are defined at the first level of sets).

If only that were what cardinality actually was, you'd be all set.

In that case my (3) definition does not hold under your arbitrary limitations.

I provided no limitations, arbitrary or otherwise.

Very simple: {} is an existing thing but its Cardinality = 0.

By using Direct Perception we immediately understand that the existence of {} is stronger than the existence of the measured members that are related to it.

Continuing to use the term isn't defining it. Please define stronger.

 

[jsfisher]

|{}|=0

|{{}}|= 0+1

|{{{}}}| = 0+1+1

...

|{{a,b,c,…}}| is |N|=|{a,b,c,…}| + 1


...

Is this all supposed to be making some important point, or do you just enjoy writing things that are incorrect?

 

[The Man]

Jsfisher, as you know I understand Cardinality different than you.

For me Cardinality is the measurement of the existence of things, and this measurement is not limited to any particular level of the measured thing (in your case Cardinality is the measurement of the existing sets that are defined at the first level of sets).

In that case my (3) definition does not hold under your arbitrary limitations.

Still haven’t figured out what “arbitrary” means have you.

Let us examine it more precisely.



Very simple: {} is an existing thing but its Cardinality = 0.

By using Direct Perception we immediately understand that the existence of {} is stronger than the existence of the measured members that are related to it. In this case no member is related to {} (including itself) and yet it exists independently of the luck of members. So there is a difference between being Membership and being Member, where the existence of Membership is stronger than or independent of the existence of members.

Well wait a second you just claimed above that “Cardinality is the measurement of the existence of things” for you. So the “existence of {}” is its “Cardinality” and can not “exists independently of the luck of members”. You still haven’t figured out what self consistency is yet either.

I call Membership Non-locality and Member Locality, and also show that no amount of Localities is Non-locality.

What you call them is irrelevant to what they actually mean. Again simply claiming does not “show” as you like to claim.

This fact (The independent existence of Membership of any Member(s)) is used AND ignored by Standard Math, which is a contradiction that is avoided by OM.

Utter nonsense as is your claim to use “Cardinality is the measurement of the existence of things” than asserting “yet it exists independently of the luck of members” the contradictions are all yours and the basis of your “OM”

Because of this -used AND ignored Standard Math contradiction- things like {{a,b,c,…}} and {a,b,c,…} are not fully understood, for example:

Cardinality 1=|{{a,b,c,…}}| is based on the existence of {a,b,c,…} by ignoring the understanding of the building-blocks that enable this existence, in the first place.

As a result the difference of Complexity between {{x}} and {x} is not learned as a significant property of the researched subjects, and things are trivialized into member's amounts without the notion of how their existence is possible, in the first place.

If one does not understand the Building-Blocks of X, one does not understand X, and therefore does not understand any full result of X.

Let us demonstrate it by using {{a,b,c,…}}.

The 1 of 1=|{{a,b,c,..}}| is possible in the first place exactly because the measured object is the result of the linkage between different levels of existence, symbolized as "{""}" and "a,b,c,…", where the existence of "{""}" is stronger than the existence of "a,b,c,…" (as clearly demonstrated by {} case).

It does not matter what symbols are used, for example:

"A" of {A} is equivalent to the internal "{""}" of {{a,b,c,…}}.

Cantor, by using exactly the fact that <--> (mapping) existence is stronger than the existence of the mapped objects (<--> exists even if there is no input\output, exactly as {} exists without members) uses this fact in order to show that he enables to define a member of the power set of X that is not mapped with any member of set X, which enables to conclude that |P(X)| > |X|.

This result is based on measuring only the first level of sets (by using AND ignoring what actually enables this measurement, which is not less than Membership(Non-locality)\Members(Localities) Linkage) and any other measurement that is not based on the first level, is arbitrarily ignored.

Since I use Direct Perception, I get sets Cardinality by not ignoring their Complexity as seen from Membership(Non-locality)\Members(Localities) Linkage.

From this "X-ray"-like comprehensive point of view, let us re-examine the following:

1) X is a set and any member of X (if exists) is a set.

2) If X is an infinite set, then |X| is a transfinite cardinal.

3) If |X| is a transfinite cardinal, such that |X| > the cardinality of any member of X and any member of X is a finite set, then |X| is the smallest transfinite cardinal.

It is shown by jsfisher that (3) does not hold if Cardinality is a measurement that is limited only to the first level of sets.

But (3) holds if Cardinality is the measurement of the existence of anything that is the result of Membership(Non-locality)\Members(Localities) Linkage.

In that case the Cardinality of X={{a,b,c,…}} is |N|=|{a,b,c,…}| + 1 and (3) holds.


In other words jsfisher "You win the battle but lose the war" exactly because you are using a reasoning that is based on arbitrary limitations.

Doron your “"X-ray"-like comprehensive point of view” is burning a hole in your assertions. Your purported contradictions are entirely yours. The application of cardinality is not “arbitrary” but expressly defined. You simply choose to make it arbitrary and meaningless by claiming it is your “measurement of the existence of things” like the empty set and then asserting “yet it exists independently of the luck of members”. You will continue to lose both the ‘battles’ and the “war”, Doron, until you can display at least some semblance of self consistency in your assertion and notions.

 

[doronshadmi]

Well wait a second you just claimed above that “Cardinality is the measurement of the existence of things” for you. So the “existence of {}” is its “Cardinality” and can not “exists independently of the luck of members”. You still haven’t figured out what self consistency is yet either.

You miss it. The Cardinality of the existence of the empty set is stronger than the cardinality of the existence of its members exactly as the Cardinality of the existence of any set is stronger than the existence of its members. Your Cardinality is the measurement of the existence of the first level of some set.

My comprehensive Cardinality is not your arbitrary and limited Cardinality.

Look how your arbitrary limitation prevents from you to get this simple fact:

{} is an existing thing but its Cardinality = 0.

In other words, the Cardinality of the existence of the empty set is stronger than the Cardinality of the existence of its members.

We do not count the Cardinality of the existence of a set as one of its members, exactly as Membership is stronger than any collection of members, and {} simply demonstrates it .

This notion is beyond your limitations, but this is your limiting reasoning, not mine.

 

[jsfisher]

You miss it. the Cardinality of the existence of the empty set is stronger than the cardinality of the existence of its members exactly as the Cardinality of the existence of any set is stronger than the existence of its members. Your Cardinality is the measurement of the existence of the first level of some set.

No, you missed it.

The term, cardinality, means something. The fact you don't understand its meaning doesn't give you leave to make up your own meaning.

By the way, I am still waiting for your definition of stronger. Got one?