Deeper than primes

AvalonXQ said:

This is math. I don't have to prove that Aleph0 exists; I just have to define what I mean when I say "Aleph0". I define Aleph0 as the cardinality of N.
Are you arguing that N has no cardinality, or are you arguing that the cardinality of N is not greater than all n in N?

Click to expand...

What you actually say is this:

Math is a science that can use an arbitrary determination that asserts that there is a quantity, which is larger than any finite quantity , and this larger quantity is somehow used as a measurement unite of the collection of finite quantities, where this collection has no the largest finite quantity.

As a result any arithmetical operation between the larger quantity and any finite quantity has no meaning since Aleph0 + or * or ^ (finite quantity)=Aleph0.

For example: the sum of the addition of all finitely many finite quantities is larger than any one of the summed finite quantities.

The sum of the addition of all infinitely many finite quantities is larger than any one of the summed finite quantities, but since the collection of infinitely many finite quantities has no largest quantity, it is not clear at all what is the exact meaning of the assertion “all infinitely many finite quantities”. Since this assertion is not clear, there is no guarantee that there is a quantity that is larger than “all infinitely many finite quantities” .

Furthermore, it is not clear what is the meaning of “The sum of the addition of all infinitely many finite quantities” or “The sum of the multiplication of all infinitely many finite quantities” or “The sum of the exponentiation of all infinitely many finite quantities” because +,* and ^ operations are indistinguishable by Cantor’s transfinite notion.

Since +,* and ^ are indistinguishable they can’t be used in order to conclude any meaningful thing according to the reasoning of Cantor’s transfinite notions .

Since this is the case Cantor’s transfinite universe is nothing but a belief that tries to use things that have a meaning only at the finite level, by forcing them on his belief about the infinite, but as can be clearly seen, it is nothing but forcing beliefs without any concrete reasoning at its basis.

doronshadmi said:

Since this is the case Cantor’s transfinite universe is nothing but a belief that tries to use things that have a meaning only at the finite level, by forcing them on his belief about the infinite, but as can be clearly seen, it is nothing but forcing beliefs without any concrete reasoning at its basis.

Click to expand...

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cantor first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality: three.

Click to expand...

And so now when we know that "cardinality" means size and that the word doesn't relate to the institution of high clergy having impact on Cantor's "belief", it's easy to evaluate A = {10111} and B = {11011}: A = B = 5 -- but only in quantitative sense. A and B maybe qualitatively different due to the position of the single space. In this view, is A consequence of B, or B consequence of A; what came first -- chicken or egg?

The Man said:

Again not only is that circular, that you simply assume your conclusion about collection (that it is "enabled" by your "atomic building-blocks" "in the first place"), to establish that your conclusion is not "based on the notion of Collection", and contradictory ("building-blocks that enable Collection, in the first place" is based on your notion of collection as your "complex"), but it is also irrelevant (by your own assertion) as you claim that your conclusion about collection is not "based on the notion of Collection"

Click to expand...

The Man said:

I have done neither. A “complex” is your notion Doron so the only one who can be using “a complex in order to define a complex” is simply you. Again you’re the one turning equal and not equal signs into your own circular symbols, clearly demonstrating your own understanding of the circularity inherent in your own notions.

Click to expand...

The Man,

By your flat reasoning anything is defined at the same level.

As a result you have no hierarchy of dependency, where hierarchy of dependency is not circular.

OM’s hierarchy of dependency goes like this.

The atomic state is the source of any manifestation.

The first level of manifestation is the extreme atomic aspects, such that no aspect is the manifestation of the other aspect, and as a result any aspect is independent of the other aspect. The linkage of the extreme manifestations is possible because they are direct manifestations of the atomic state.

These direct manifestations depends of the atomic state but the atomic state does not depend on its manifestations.

At the manifestation level, each atomic aspect is not the manifestation of the other aspect, and as a result we get a complex level, which is not any of its atomic aspects alone.

0 < (∞\0) < ∞ is exactly the realm that is the manifestation of the atomic state (Level 0), where the manifestation has two levels:

Leve1) Each extreme atomic aspect, notated as 0 or ∞, that is based on Level 0, such that no aspect is the manifestation of the other aspect

Level2) The (∞\0) complex manifestation that is based on Level 1, where each atomic aspect is not the manifestation of the other aspect under (∞\0) complex manifestation.

No flat reasoning can get that, as clearly seen in your case, The Man.

epix said:

And so now when we know that "cardinality" means size and that the word doesn't relate to the institution of high clergy having impact on Cantor's "belief", it's easy to evaluate A = {10111} and B = {11011}: A = B = 5 -- but only in quantitative sense. A and B maybe qualitatively different due to the position of the single space. In this view, is A consequence of B, or B consequence of A; what came first -- chicken or egg?

Click to expand...

You still do not notice that you are using non-local things like "=" or the outer "{" and "}" and local things like "A","B","0","1","5", which provide a researchable framework only if they are simultaneously used as Relation ("=" or the outer "{" and "}") \ Element ("A","B","0","1","5") Interaction (REI).

There is no serial "chicken or egg" case here, because REI is simultaneous.

epix said:

And so now when we know that "cardinality" means size

Click to expand...

So you are using the word "size" instead of "infinite quantity".

Still a concrete reasoning of "infinite quantity" is not provided, and Cantor's notions are nothing but a belief.

jsfisher said:

More utter nonsense. Definitions do not require proof. For that matter, a proof that the cardinality of the set of integers is larger than any finite cardinal number is independent of what we may choose to call the cardinality of the set of integers.

Doron, you are imagining circles. You are seeing things that are just not there. Give it up. You impaired vision is no substitute for comprehension.


Watch: Let aleph-0 be the cardinality of the set of integers.

There. Done. No circles.

Click to expand...

http://www.internationalskeptics.com/forums/showpost.php?p=5457129&postcount=7721

Apathia said:

But of course, Doron.
Aleph0 is a collection, and it's not Absolute Infinity.
It's a member of a special class of elements, local infinities, that are REIs resulting from Absolute Infinite and Absolute Finite Linkage.

Cantor called them "tranfinites," because he understood they were not Absolutely Infinite.


(bolding mine)

Consider that there is a way to reconcile, to see a non-local integration, of mathematical infinity under your OM framework.
The direction of Organic Thinking is toward integration not local-only exclusion.

(Never mind Cantor's "Deo." I understand that the "Divine" in your context is best represented by the "Singularity," or the Trunk.)

Click to expand...

Cantor left Non-locality out of his research about Infinity, and as a result his system is closed under the concept of Collection of localities, without understanding the concept of Collection, in the first place.

Furthermore, he did not understand the actual-finite (totally local) and the actual-infinite (totally non-local) as the independent building-blocks that enable Collection.

Also he did not understand that Collection is a complex, such that 0(actual-finite) < (∞\0 complex) < ∞(actual-infinite), which means that any complex is incomplete w.r.t 0 or ∞.

By understanding that 0(actual-finite) < (∞\0 complex) < ∞(actual-infinite), the notion of transfinite cardinality is collapsed because any complex is incomplete w.r.t the actual.

In other words, the class of infinite cardinals does not hold water.

doronshadmi said:

Cantor left Non-locality out of his research about Infinity, and as a result his system is closed under the concept of Collection of localities, without understanding the concept of Collection, in the first place.

Furthermore, he did not understand the actual-finite (totally local) and the actual-infinite (totally non-local) as the independent building-blocks that enable Collection.

Also he did not understand that Collection is a complex, such that 0(actual-finite) < (∞\0 complex) < ∞(actual-infinite), which means that any complex is incomplete w.r.t 0 or ∞.

By understanding that 0(actual-finite) < (∞\0 complex) < ∞(actual-infinite), the notion of transfinite cardinality is collapsed because any complex is incomplete w.r.t the actual.

In other words, the class of infinite cardinals does not hold water.

Click to expand...

1) You can add "closed" to the list of terms that do no mean what you think they do.

2) if you still refuse to learn proper maths, at least take up philosophy formally, it will help you in organizing your thoughts and assembling your arguments.

sympathic said:

1) You can add "closed" to the list of terms that do no mean what you think they do.

2) if you still refuse to learn proper maths, at least take up philosophy formally, it will help you in organizing your thoughts and assembling your arguments.

Click to expand...

You can add Complexity and Collection to the list of terms that do no mean what you think they do.

doronshadmi said:

You can add Complexity and Collection to the list of terms that do no mean what you think they do.

Click to expand...

You are wrong. These are defined and well accepted terms. If you insist on using the exact same terms for something else (something you have been told many times that is not a recommended thing to do) you should provide proper definitions for them. You have failed to do so.

doronshadmi said:

You can add Complexity and Collection to the list of terms that do no mean what you think they do.

Click to expand...

If you are referring to the meanings you've assigned to those words, then that is entirely due to your inability to communicate those meanings. That is assuming you actually have concise and consistent meanings in mind.

doronshadmi said:

You can add Complexity and Collection to the list of terms that do no mean what you think they do.

Click to expand...

And by the way - please stop these projections. They are annoying and counter productive.

Natural numbers are used as ordinals and cardinals.

But in both cases we find infinitely many locations along an endless path, which their quantity is incomplete w.r.t to it:
no matter how they are packed, summed (by +,* or ^) or arranged along the endless path.

Cantor did not understand this simple fact, which actually eliminates the reasoning of his transfinite system.

sympathic said:

You are wrong. These are defined and well accepted terms. If you insist on using the exact same terms for something else (something you have been told many times that is not a recommended thing to do) you should provide proper definitions for them. You have failed to do so.

Click to expand...

sympathic,

You have failed to understand the right meaning (which is not the currently agreed definitions) of these concepts.

Your definitions do not hold water because they have no basis (complex is used to define complex).

doronshadmi said:

Cantor left Non-locality out of his research about Infinity, and as a result his system is closed under the concept of Collection of localities, without understanding the concept of Collection, in the first place.

Furthermore, he did not understand the actual-finite (totally local) and the actual-infinite (totally non-local) as the independent building-blocks that enable Collection.

Also he did not understand that Collection is a complex, such that 0(actual-finite) < (∞\0 complex) < ∞(actual-infinite), which means that any complex is incomplete w.r.t 0 or ∞.

By understanding that 0(actual-finite) < (∞\0 complex) < ∞(actual-infinite), the notion of transfinite cardinality is collapsed because any complex is incomplete w.r.t the actual.

In other words, the class of infinite cardinals does not hold water.

Click to expand...

There is a simple way of parsing the matter, that would spare you the grief of having to throw almost all of mathematics in the dumpster.
Rocognize that there is an absolute mode of speaking and a realtive mode of speaking.
For example: the set of all numbers which are the square root of nine is {3} which is relatively speaking complete.
Absolutely speaking it is not Complete.
You can say "local-only" is is complete with 3.
And leave that "local-only" calculation for what it's worth to bankers.
It woul'd be very scary if they declared that Organic Mathematics said my payment wasn't "Complete" and continued to bill me.

You don't have to dissolve ordinary mathematics to get to your spiritual goals.
When it comes to those, you speak in an Absolute or matamathmagical mode.
That's where you work with the concepts born if the interaction of Absolute Infinity and Absolute Finitude.
You can let Cantor haver his "Tranfinites" while you persue the Absolute Reality.

Ther's no sin in manipulating and caculating locally when the context calls for that.
You sinply want such activities to be agknowleged as contained in a higher reality

When a sphere passes through the metaphorical Flatland, it appears as a growing then dismishing circle. The Flatlanders see only circles, unless they get a view from "above."
The passing sphere does not invalidate circles.
It does not preach that circles are false concepts.
It instead snatches a researcher out of his 2d world, so he can understand what those circles are about.

Imagine a Flatland geometer who because he has witnessed the "Intersection of The Sphere," concludes that there are no real circles, just semicircles.
He's confused about what he saw and its application to Geometry.

Your work is about Infinity in the Absolute sense. And you deny that there could be any kind of relative infinite (or transfinite) as Cantor put it.
Perhaps you find such a concept obscene or a kind of profanity. The same for "completeness." Only the absolute sense is acceptable to you.

But don't you catch it that in your own framework Absolute Finitude makers its contribution creating the relative sense of things: a local completeness, even a local infinity (of the relatve sort)?
The Non-Local and the Local combine in very interesting ways.

Nevermind Cantor. You point out the contribution the Non-Local or Absolute Infinity makes to mathematical concepts. Don't ignore the place of the Local Atom or Absolute Finitude to create the useful relative and local calculation.

Let the circles be whole.

doronshadmi said:

sympathic,

You have failed to understand the right meaning (which is not the currently agreed definitions) of these concepts.

Your definitions do not hold water because they have no basis (complex is used to define complex).

Click to expand...

Well, at least you did not use projections. Thanks.

doronshadmi said:

Natural numbers are used as ordinals and cardinals.

Code:

[U]Ordinals:[/U]

1 2 3 4 5 6 7 8 9  ...
[u]. . . . . . . . .       [/u]...

The symbols represent infinitely many ordered positions along a path.



[U]Cardinals:[/U]

 1    2      3   ...
[u](.) (. .) (. . .)       [/u]...

The symbols represent infinitely many finite quantities 
(order has no significance) along a path.

But in both cases we find infinitely many locations along an endless path, which their quantity is incomplete w.r.t to it:

Code:

[u]. . . . . . . . . . . . . . [/u] ...

no matter how they are packed, summed (by +,* or ^) or arranged along the endless path.

Cantor did not understand this simple fact, which actually eliminates the reasoning of his transfinite system.

Click to expand...

Here (and most anywhere else) you use the Line as a metaphor for Absolute Infinity.
Cantor's line was the traditional mathematical object, relative instead of absolute.
Pehaps we could allow him the distinction he made between Absolute and mathematical infinities.

Apathia said:

Your work is about Infinity in the Absolute sense. And you deny that there could be any kind of relative infinite (or transfinite) as Cantor put it.

Click to expand...

On the contrary. Because I deal with Infinity in the Absolute sense I have the ability to understand Infinity in the relative (potential) sense, which is exactly the inherent incompleteness of any complex (and a collection is a form of a complex) w.r.t Infinity in the Absolute sense.

Apathia said:

Here (and most anywhere else) you use the Line as a metaphor for Absolute Infinity.
Cantor's line was the traditional mathematical object, relative instead of absolute.
Pehaps we could allow him the distinction he made between Absolute and mathematical infinities.

Click to expand...

It cannot be done since transfinite cardinality holds only if there is such a thing like a complete infinite collection.

Since such a thing is inherently impossible, then Cantor's transfinite system does not hold.

Apathia said:

You don't have to dissolve ordinary mathematics to get to your spiritual goals.

Click to expand...

Call my goals whtever you like, they can't be acheived as long as the real nature of Complexity is not understood, and it is not understood exactly because of notions like the Cantorean notion about complete infinite collections.

I define Aleph0 as the cardinality of N.
Are you arguing that N has no cardinality, or are you arguing that the cardinality of N is not greater than all n in N?

AvalonXQ said:

I define Aleph0 as the cardinality of N.
Are you arguing that N has no cardinality, or are you arguing that the cardinality of N is not greater than all n in N?

Click to expand...

At the moment that you understand that any infinite complex is inherently incomplete (it is not absolute infinity) you also understand that no infinite complex (and an infinite collection is a form of an infinite complex) has an accurate cardinality.


EDIT: For better understanding, please see http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732.

doronshadmi said:

At the moment that you understand that any infinite complex is inherently incomplete (it is not absolute infinity) you also understand that no infinite complex (and an infinite collection is a form of an infinite complex) has an accurate cardinality.

Click to expand...

Ah. So you're arguing that N has no cardinality.

AvalonXQ said:

Ah. So you're arguing that N has no cardinality.

Click to expand...

It has no accurate cardinality in the sense of a finite cardinality.



EDIT: Please see http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732.

You mean N is an infinite set? True, but that's hardly an earth-shocking discovery.

Skeptic said:

You mean N is an infinite set? True, but that's hardly an earth-shocking discovery.

Click to expand...

http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732.

doronshadmi said:

It has no accurate cardinality in the sense of a finite cardinality.

Click to expand...

This is true. However, you have not proven that a cardinality must be finite in order to be accurate.
Cardinality is defined in terms of the existence of a bijective function.
N has the cardinality of Aleph0 (by definition), and then many other sets have that same cardinality, including Q, because there is a bijection between them and N. This is a well-defined demonstrable mathematical concept; it's not faith-based at all.

AvalonXQ said:

This is true. However, you have not proven that a cardinality must be finite in order to be accurate.
Cardinality is defined in terms of the existence of a bijective function.
N has the cardinality of Aleph0 (by definition), and then many other sets have that same cardinality, including R, because there is a bijection between them and N. This is a well-defined demonstrable mathematical concept; it's not faith-based at all.

Click to expand...

Why do you ignore http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732 ?


Bijective function says nothing about the completeness of the mapped collections.

doronshadmi said:

The Man,

By your flat reasoning anything is defined at the same level.

Click to expand...

Another label Doron, you seem to manifest them as much as you do everything else.

doronshadmi said:

As a result you have no hierarchy of dependency, where hierarchy of dependency is not circular.

Click to expand...

Doron, you are the one making equal and not equal signs circular to demonstrate the circular aspects at the foundation of your notions.

doronshadmi said:

OM’s hierarchy of dependency goes like this.

The atomic state is the source of any manifestation.

Click to expand...

Apparently not since you just manifested your contrivance of “The atomic state”.

doronshadmi said:

The first level of manifestation is the extreme atomic aspects, such that no aspect is the manifestation of the other aspect, and as a result any aspect is independent of the other aspect. The linkage of the extreme manifestations is possible because they are direct manifestations of the atomic state.

Click to expand...

Are you sure you couldn’t have stuck the “manifestation” in there a few more times?

This is your second manifestation, not your first and it is simply a contrivance as your “atomic state” is.

doronshadmi said:

These direct manifestations depends of the atomic state but the atomic state does not depend on its manifestations.

Click to expand...

So are you now claiming that your “extreme atomic aspects” are not aspects of your “atomic state” as you claimed before?

doronshadmi said:

At the manifestation level, each atomic aspect is not the manifestation of the other aspect, and as a result we get a complex level, which is not any of its atomic aspects alone.

0 < (∞\0) < ∞ is exactly the realm that is the manifestation of the atomic state (Level 0), where the manifestation has two levels:

Click to expand...

You just love that word “manifestation”, don’t you? Understandable since you apparently enjoy simply manifesting things to suit your contrivances.

doronshadmi said:

Leve1) Each extreme atomic aspect, notated as 0 or ∞, that is based on Level 0, such that no aspect is the manifestation of the other aspect

Level2) The (∞\0) complex manifestation that is based on Level 1, where each atomic aspect is not the manifestation of the other aspect under (∞\0) complex manifestation.

Click to expand...

Doron they are all just manifestations of your imagination and serve only to feed your imagination that you have created something actually useful.

doronshadmi said:

No flat reasoning can get that, as clearly seen in your case, The Man.

Click to expand...

Doron no manifestations of your imaginary reasoning will be as consistent, coherent and useful as actual math and you simply can’t get that, as clearly seen in your case.

AvalonXQ said:

N has the cardinality of Aleph0 (by definition), and then many other sets have that same cardinality, including R, because there is a bijection between them and N. This is a well-defined demonstrable mathematical concept; it's not faith-based at all.

Click to expand...

You probably mean Q (the rationals), not R (the reals).

The (∞\0) complex manifestation that is based on Level 1, where each atomic aspect is not the manifestation of the other aspect under (∞\0) complex manifestation.

Click to expand...

This means absolutely nothing at all.

doronshadmi said:

http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732.

Click to expand...

Ah, yes. So Cantor is now another one of those idiots who just didn't understand mathematics.

doronshadmi said:

Why do you ignore http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732 ?

Click to expand...

I did not ignore it. I will re-state my reasoning.
Cardinality is defined as a property of a set such that two sets have the same cardinality if and only if a bijection exists between them.
Because the identity function acts as a bijection from a set onto itself, every set (finite or infinite) has the same cardinality as itself.
We define Aleph0 to represent the cardinality of N, and then we see that many other sets (including Q) also have cardinality Aleph0.
This reasoning is not flawed. If you consider it to have a flaw, please identify exactly where the flaw is.

ETA:

Skeptic said:

You probably mean Q (the rationals), not R (the reals).

Click to expand...

Yes, thank you.

The Man said:

Another label Doron, you seem to manifest them as much as you do everything else.



Doron, you are the one making equal and not equal signs circular to demonstrate the circular aspects at the foundation of your notions.



Apparently not since you just manifested your contrivance of “The atomic state”.



Are you sure you couldn’t have stuck the “manifestation” in there a few more times?

This is your second manifestation, not your first and it is simply a contrivance as your “atomic state” is.



So are you now claiming that your “extreme atomic aspects” are not aspects of your “atomic state” as you claimed before?



You just love that word “manifestation”, don’t you? Understandable since you apparently enjoy simply manifesting things to suit your contrivances.



Doron they are all just manifestations of your imagination and serve only to feed your imagination that you have created something actually useful.



Doron no manifestations of your imaginary reasoning will be as consistent, coherent and useful as actual math and you simply can’t get that, as clearly seen in your case.

Click to expand...

Another demonstration of flat reasoning.

How boring.

AvalonXQ said:

Because the identity function acts as a bijection from a set onto itself, every set (finite or infinite) has the same cardinality as itself.

Click to expand...

Great.

And in the case of an infinite collection the identity function has the property of the inherent incompleteness of the mapped collection.

Skeptic said:

Ah, yes. So Cantor is now another one of those idiots who just didn't understand mathematics.

Click to expand...

It is a matter of understanding the subject.

The personal attitude has nothing to do with the re-searched subject.

doronshadmi said:

And in the case of an infinite collection the identity function has the property of the inherent incompleteness of the mapped collection.

Click to expand...

No. The bijection is complete -- it maps every element onto itself.
The function f=n for all n in N completely maps N. If you disagree, simply tell me what element in N it misses.

doronshadmi said:

On the contrary.

Click to expand...

Here's a brief chart of positions:

Potential Infinitiesl Mathmatical Transfinitesl Absolute "ontological"
Infinity

Cantorl Cantorl Cantor
Doronl Doron
Apathial Apathial

Did you mean:

doronshadmi said:

Another demonstration of flat reasoning.

How boring.

Click to expand...

Another evasion by labeling

How typical.

zooterkin said:

Did you mean:

Potential Infinities| Mathematical Transfinites| Absolute "ontological" Infinity
Cantor| Cantor| Cantor
Doron| | Doron
Apathia| Apathia|

Click to expand...

Yes!
Thank you, Zooterkin.

doronshadmi said:

http://www.internationalskeptics.com/forums/showpost.php?p=5457129&postcount=7721

Click to expand...

Yes, and that post is utter nonsense, too. Therein you invented the following bit of ignorance:

doronshadmi said:

Math is a science that can use an arbitrary determination that asserts that there is a quantity, which is larger than any finite quantity , and this larger quantity is somehow used as a measurement unite of the collection of finite quantities, where this collection has no the largest finite quantity.

Click to expand...

No. Absolutely not.

Saying, "Let aleph-0 be the cardinality of the set of integers," is simply a definition, nothing more. That let's us say, "aleph-0", which is a lot more compact and convenient than having to say, "the cardinality of the set of integers," whenever we need to refer to that particular thing.

Everything you wrote is a grossly incorrect contrivance, oblivious to the actual reality of what is a definition.