Deeper than primes

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doronshadmi said:
Cantor did not understand this simple fact, which actually eliminates the reasoning of his transfinite system.
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Repeating your incorrect statement does not render in correct. Even an infinite repitition of false remains false.
 
doronshadmi said:
So you are using the word "size" instead of "infinite quantity".
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I adopted the word "size" from the article the part of which I quoted from and didn't see any reason to change the terminology, coz the subject was more or less living in the Naive set theory.
http://en.wikipedia.org/wiki/Naive_set_theory

But in more formal and rigorous environment, the word "size" in connection with braces, which bind finite sets, may cause a problem. Since braces are used to accommodate elements of the set, they can be compared to the size of apparel: {}, {_}, {__}, {___}, {____}, and larger.

But the usage of braces cannot determine the Cardinal Set with respect to its size and content. For that purpose, you need to switch to parentheses. First, you create a verbal and a symbolic link.

parentheses: ()

The size, or the width, of () is determined by a subset in the verbal part, which becomes non-local:

--re--he-es: (pants)

So with proper syntax and symbolism, you can handle hard-to-solve problem such as A = (tuw...xyz), where the lower-case letters stand for uknown elements of set A.
 
zooterkin said:
You're welcome; I can only claim to understand how to format the table, not the contents.
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Philosophy never lets anything be dirt simple.
There are different types, aspects, or flavors of infinity.

There's "potential" infinity which is just to say that there is a tendency to go on increasing or decreasing with infinity as the destination referenced but never reached.

There's mathematical infinity that deals in sets of infinite content, some countable, some uncountable.

And there's Infinity regarded as an existing transcendent reality beyond any completion or containment.

Not mentioned in my post is physical infinity, notions of infinite time or space as a reality in science.

No one has a quarrel with potential infinity. Aristotle preferred it above any other concept of the infinite, because it doesn't entail any of those pesky paradoxes.

The manipulation of completed infinities in convergences and limits, and more recently in sets is a modern (relatively speaking) addition to Mathematics.
Cantor addressed the paradoxes via his "Transfinite Numbers."
So few now question infinity's place in Mathematics.

Absolute Infinity regards Infinity as metaphysically existing prior to any concept of the infinite and being unreachable, unobtainable, and beyond containment in any set.

So enter Doron:
He has no problem with potential infinity. It's merely a composite concept.
For him the actual infinity is the Absolute Infinity.
He regards mathematical infinity as a false concept, for the Real thing is the ever transcendent absolute.

I suggested he might consider mathematical infinity as a legit concept arising from the interaction of his Absolute Infinity and Absolute Finitude (the Local/Non-Local combo).

He doesn't buy that for at least two reasons:
One, he regards potential infinity as being that composite concept.
Two, mathematical infinity cannot be Real, because transfinites cannot claim to be infinite any any way. Only the pure, unsullied, wholly other, Absolute Infinity has the claim to Reality. Anything that can be packaged in a set falls short of absolute transcendence.

Now, myself, heretical to both Doron and Cantor, regard the Absolute Infinite as merely a conceptual projection. Transcendence, as I see it, is merely an open vastness empty of any ultimate, metaphysical content.
I get this from the philosophical tradition of Mahayana Buddhism.
But I'm not dogmatic about it.
It's metaphysics,
and I refuse to get worked up over metaphysical questions.

I'm content to let Doron have his approach that has fascinating dovetails with Vedic conceptions of the Divine.
In my pragmatism, though, I sought a way he might advance his structure without having to throw centuries of useful mathematics into the landfill.
However ...

But, as usual, we succeed in getting Doron to state his divergent views more directly where they can by evaluated head on.
 
Apathia said:
Philosophy never lets anything be dirt simple.
There are different types, aspects, or flavors of infinity.

There's "potential" infinity which is just to say that there is a tendency to go on increasing or decreasing with infinity as the destination referenced but never reached.

There's mathematical infinity that deals in sets of infinite content, some countable, some uncountable.

And there's Infinity regarded as an existing transcendent reality beyond any completion or containment.
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The term "potential infinity" has been attributed to Aristotle who decided to raise the concept of infinity from the state of "aperion" -- some archaic synonym to "chaos." So the concept of infinity must have been an abstract issue beyond easy grasp for the Greek thinkers.

There is not much known about the fate of infinity in earlier times due to illiteracy. But I don't think that an ancient culture that couldn't develop some form of written language would be able or even willing to ponder highly abstract issues, which were pretty useless for a practical application.

Since the ancient Greeks are classified Homo sapiens sapiens, as we are, the concept of infinity when explained in the aspects that open the door to calculus can be understood by high school kids, but the Greeks never came indirectly near to the concept of relativity. And that means, this strongly counterintuitive concept is very difficult to grasp without math sitting by tutoring. Einstein could do it though and could work with his hunch.
 
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epix said:
The term "potential infinity" has been attributed to Aristotle who decided to raise the concept of infinity from the state of "aperion" -- some archaic synonym to "chaos." So the concept of infinity must have been an abstract issue beyond easy grasp for the Greek thinkers.

There is not much known about the fate of infinity in earlier times due to illiteracy. But I don't think that an ancient culture that couldn't develop some form of written language would be able or even willing to ponder highly abstract issues, which were pretty useless for a practical application.

Since the ancient Greeks are classified Homo sapiens sapiens, as we are, the concept of infinity when explained in the aspects that open the door to calculus can be understood by high school kids, but the Greeks never came indirectly near to the concept of relativity. And that means, this strongly counterintuitive concept is very difficult to grasp without math sitting by tutoring. Einstein could do it though and could work with his hunch.
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Yes. The Ancient Greeks and the Ancient Hindu authors of the Vedas were certainly every bit Homo Sapiens Sapiens as ourselves when it comes to pondering abstract concepts.

Aristotle preferred infinity as a potentiality as oppsed to an actual infinty which seemed chaotic and irrational.
Even many contemporary educated people (I've given my ex-girlfriend as an example.) find infinity confusing and counterintuitive, just as they find Realtivity and Quantum Theory to be weird stuff.

Though my xgf had taken Calculus in the course of her degree in Biology,
she found it made more sense to her that a decimal point followed by an infinite number of 9s (.999999999.......) never comes to equaling one. Common sense says it's never going to make it there.

If you regard Infinity asd a metaphysical transcendent, then it just naturally follkows that it is a fundamental entity that cannot be obtained, divided, or contained. Thus Doron's objection to mathematical infinity which treats the infinite as a completed collection.
It seems Cantor recognized that and classed his infinities as a variety of finites (transfinites).
But Cantor still exceeds my xgfs common sense and commits hubris as far as Doron is concerned.
(It's like claiming divine attributes for a mere mortal.)
 
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Apathia said:
If you regard Infinity asd a metaphysical transcendent, then it just naturally follkows that it is a fundamental entity that cannot be obtained, divided, or contained.
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Yes, infinity cannot be trivially manipulated, but it can be imagined. In terms of length, infinity is a distance that I like to see between myself and my mother-in-law.
 
epix said:
Yes, infinity cannot be trivially manipulated, but it can be imagined. In terms of length, infinity is a distance that I like to see between myself and my mother-in-law.
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Or, as I was explaining to my housemate yesterday, his prospective employer will never see his Micky Mouse resume, because it will be at the bottom of the pile and will never be reached, because HR will have to go through half the pile first, then half that pile, and half again and so on ad infinitim. :wackylaugh:
 
Before Doron regroups and attacks again, there is an issue worth mentioning that involves Cantor's Theorem that says, "For any set X, the power set of X is cardinally larger than X." That's the idea Doron frowns upon. But it may be true that Cantor tailored an infinity model for his own needs -- he created a special infinity willing to obey his rules.

From the wording of Cantor's theorem, "any set X" means this:

X = {a, b, c, d, e, . . .}

That means the number of different elements (none of them is identical to each other) that the letters represent approaches infinity, but by Cantor, it is also the "largest" set of all sets that happen to approach infinity at that moment. Now comes the "power set of X" with such a characteristic, which proves that the set X cannot be the largest set. The power set P(X) is created from subsets of set X according to "one on one correspondence."

P(X) = { {a }, { a, b }, { b, c, e }, { a, c }, { e }, . . . }

Cantor will use flawless logic to prove that se P(X) is larger than X. But if you examine the elements of set X, you see that each element is distinct and therefore exist only once, whereas in the power set P(X), element "a", for example, appears more than once. And that's the special condition that allows the proof.

http://www.mathacademy.com/pr/prime/articles/cantor_theorem/index.asp
 
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jsfisher said:
Repeating your incorrect statement does not render in correct. Even an infinite repitition of false remains false.
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A collection is a Complex.

No complex is complete w.r.t the actual, and it does not matter if it is a collection of False True (or whatever) statments.

A collection remains incomplete w.r.t the actual, which is a notion that your flat "closed under Complexity" reasoning can't comprehend.
 
epix said:
I adopted the word "size" from the article the part of which I quoted from and didn't see any reason to change the terminology, coz the subject was more or less living in the Naive set theory.
http://en.wikipedia.org/wiki/Naive_set_theory

But in more formal and rigorous environment, the word "size" in connection with braces, which bind finite sets, may cause a problem. Since braces are used to accommodate elements of the set, they can be compared to the size of apparel: {}, {_}, {__}, {___}, {____}, and larger.

But the usage of braces cannot determine the Cardinal Set with respect to its size and content. For that purpose, you need to switch to parentheses. First, you create a verbal and a symbolic link.

parentheses: ()

The size, or the width, of () is determined by a subset in the verbal part, which becomes non-local:

--re--he-es: (pants)

So with proper syntax and symbolism, you can handle hard-to-solve problem such as A = (tuw...xyz), where the lower-case letters stand for uknown elements of set A.
Click to expand...

Actual non-locality is (__)__ , such that given any domain ____ cannot be captured by the given domain exactly because ____ is a non-local atom.
 
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AvalonXQ said:
No. The bijection is complete -- it maps every element onto itself.
The function f(n)=n for all n in N completely maps N. If you disagree, simply tell me what element in N it misses.
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There is no such a thing like a complete infinite collection, at the moment that you understand that no amout of elements (where their minimal representation is points) can be an endless atomic line (which is the minimal representation of actual infinity).

Please this time do not ignore http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732.
 
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A couple of fine points for clarification:

epix said:
...Cantor's Theorem...says, "For any set X, the power set of X is cardinally larger than X."...From the wording of Cantor's theorem, "any set X" means this:

X = {a, b, c, d, e, . . .}
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No, actually in Cantor's theorem, any set means any set, any set at all, including finite sets. However, for finite sets, the proof is trivial so the bulk of the discussion focuses on infinite sets.

...Cantor will use flawless logic to prove that se P(X) is larger than X. But if you examine the elements of set X, you see that each element is distinct and therefore exist only once, whereas in the power set P(X), element "a", for example, appears more than once. And that's the special condition that allows the proof.
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No, the element "a" does not appear as a member of the power set at all.

By the way, and not to introduce a lengthy aside, there is no actual requirement in an axiomatic set theory like ZFC that the elements of a set be distinct. In fact, {a, a, b, b, a} is a perfectly legitimate set, and it happens to be identical to the set {a, b}. This apparent anomaly is simply a consequence of the axioms giving us no way to distinguish between and among repeated elements.
 
dafydd said:
Can someone translate that for me? I don't speak Pure Unadulterated Gibberish.
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AvalonXQ said:
Me neither. We were doing so well until doron stopped speaking math and started speaking doronese.
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Doron has a personal issue with infinite sets. He cannot accept the idea of the set of all integers, for example, since a set cannot possibly contain all of them. Or so he thinks. You may have seen him get wrapped around an axle over the phrasing, "for all x in S". Same deal.

In Cantor's time, there were several accepted models for infinity. None of the various proponents were as militant as our DoronShadmi, but there was some heated debate over which model as the most useful / elegant / whatever.

For Cantor's treatment of the transfinite numbers, he needed to be explicit about his model for infinite sets. He required his infinite sets be "completed," basically meaning he could have sets of all somethings, even when there were infinitely many somethings.

DoronShadmi, being a one-track thinker, does not understand that Mathematics is perfectly capable of encompassing multiple, even contradictory models. Whereas the rest of the world has accepted that Cantor's view is most useful / elegant / whatever and mostly abandoned the other models, DoronShadmi is steadfast. But he cannot be happy merely promoting is very own inconsistent, self-contradictory model, he demands it be at the exclusion of all others.

And so he will oft repeat nonsense like "there is no such a thing like a complete infinite collection. [sic]" For through repetition, he makes it true and shows Cantor to be a fool.
 
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jsfisher said:
Doron has a personal issue with infinite sets. He cannot accept the idea of the set of all integers, for example, since a set cannot possibly contain all of them. Or so he thinks. You may have seen him get wrapped around an axle over the phrasing, "for all x in S". Same deal.

In Cantor's time, there were several accepted models for infinity. None of the various proponents were as militant as our DoronShadmi, but there was some heated debate over which model as the most useful / elegant / whatever.

For Cantor's treatment of the transfinite numbers, he needed to be explicit about his model for infinite sets. He required his infinite sets be "completed," basically meaning he could have sets of all somethings, even when there were infinitely many somethings.

DoronShadmi, being a one-track thinker, does not understand that Mathematics is perfectly capable of encompassing multiple, even contradictory models. Whereas the rest of the world has accepted that Cantor's view is most useful / elegant / whatever and mostly abandoned the other models, DoronShadmi is steadfast. But he cannot be happy merely promoting is very own inconsistent, self-contradictory model, he demands it be at the exclusion of all others.

And so he will oft repeat nonsense like "there is no such a thing like a complete infinite collection. [sic]" For through repetition, he makes it true and shows Cantor to be a fool.
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Thank you for that concise explanation,and no spectacles involved! The idea that the set of all integers contains all the said integers seems obvious even to a layman like me. If I have to decide who the fool is,Cantor or Doron you can probably guess what my decision would be.
 
dafydd said:
Thank you for that concise explanation,and no spectacles involved! The idea that the set of all integers contains all the said integers seems obvious even to a layman like me. If I have to decide who the fool is,Cantor or Doron you can probably guess what my decision would be.
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In fairness to Doron, some of the concepts he seems to express aren't totally off the wall. They don't conform to the modern view, but that doesn't mean they are without merit.

Doron's real failings are (1) his inability to accept that Mathematics can encompass more than one model and (2) general muddle from a general lack of reasoning ability and attention to consistency. The exclusive world he's created is dominated by undefined terms, circular reasoning, and blatant contradiction.
 
jsfisher said:
DoronShadmi, being a one-track thinker, does not understand that Mathematics is perfectly capable of encompassing multiple, even contradictory models.
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Some correction.

Jsfisher contradict his own words by not enable to comprehend the difference between a complex and atomic states. As a result he can't understand that Cantor's "most useful / elegant / whatever" model of the infinite is based on circular reasoning exactly because his model concludes things about Complexity, by using Complexity as the premise.

This circular reasoning is a direct result of the lack of a research of the fundamentals that enable Complexity, in the first place.

In other words, Contemporary Mathematics uses Cantor's model without understand the fundamentals that enables it, in the first place.


EDIT:

Furthermore.

The current paradigm of the mathematical science is based on Deduction, where deductive systems are context dependent and disjoint frameworks.

So the claim that the mathematical science as a whole enables models that contradict each other, is ridicules, because by the current mathematical paradigm there is no common basis to these models (here you are using a comparison ability, which is used by you as a hidden-assumption, which enables you to conclude something about A AND B deductive models. But this comparison ability has no model under your Deductive-only paradigm, so the phrase "A AND B models contradict each other" has no basis by your Deductive-only paradigm).

Actually the phrase "mathematical branches" is a false phrase under the current paradigm, because there is no "trunk" (common basis) to these "branches".

Organic Mathematics fundamentally changes this "pure" deductive context-dependent fragmented disjoint pieces of knowledge, into a one organic framework, where the phrase "mathematical branches" is a True phrase, because there is a "trunk" (common basis) to these "branches".

jsfisher, the Deductive-only mathematical paradigm can't comprehend http://www.internationalskeptics.com/forums/showpost.php?p=5418456&postcount=7505.

jsfisher said:
For through repetition, he makes it true and shows Cantor to be a fool.
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There is nothing personal in my claim against Cantor's reasoning of the infinite.

At the moment that one enables to get the Atomic\Complex framework, the Cantorean Complex-only framework simply does not hold exactly as shown at http://www.internationalskeptics.com/forums/showpost.php?p=5457647&postcount=7732.
 
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dafydd said:
Thank you for that concise explanation,and no spectacles involved! The idea that the set of all integers contains all the said integers seems obvious even to a layman like me. If I have to decide who the fool is,Cantor or Doron you can probably guess what my decision would be.
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As long as your reasoning is closed under the concept of Collection (which is a form of Complexity) things look obvious.

At the moment that you understand the difference between Complexity and Atomic, you are able to understand exactly why any collection is inherently incomplete w.r.t the atomic state.

Since you are not there (yet) you do not have the needed basis in order to comprehend OM.
 
jsfisher said:
In fairness to Doron, some of the concepts he seems to express aren't totally off the wall. They don't conform to the modern view, but that doesn't mean they are without merit.

Doron's real failings are (1) his inability to accept that Mathematics can encompass more than one model and (2) general muddle from a general lack of reasoning ability and attention to consistency. The exclusive world he's created is dominated by undefined terms, circular reasoning, and blatant contradiction.
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You are closed under Complexity without understanding its foundations.

As a result you are using a circular framework.


Whet you comprehend as inconsistent, contradiction, etc… is exactly a boomerang effect of your tries to understand Atomic\Complex framework only in terms of the Complex.
 
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doronshadmi said:
As long as your reasoning is closed under the concept of Collection (which is a form of Complexity) things look obvious.

At the moment that you understand the difference between Complexity and Atomic, you are able to understand exactly why any collection is inherently incomplete w.r.t the atomic state.

Since you are not there (yet) you do not have the needed basis in order to comprehend OM.
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My reasoning is open regarding the states of Atomic Complexity and Complex Atomicity.An inherently incomplete infinite set has both local and non-local locally non-complex complexities.Manifested complexities manifest themselves in both unitary and multi-value systems.
 
dafydd said:
My reasoning is open regarding the states of Atomic Complexity and Complex Atomicity.An inherently incomplete infinite set has both local and non-local locally non-complex complexities.Manifested complexities manifest themselves in both unitary and multi-value systems.
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Openess is exactly the inability of infinitely many . to be _____ , as clearly seen by
Code: 
[u]. . . . . . . . . . . . . . [/u] ...

It is simple exactly because Complexity is the result of __ \ . linkage
 
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doronshadmi said:
Openess is exactly the inability of infinitely many . to be _____ , as clearly seen by
Code: 
[u]. . . . . . . . . . . . . . [/u] ...

It is simple exactly because Complexity is the result of __ \ . linkage
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No, it is simply because you didn't use enough dots.
 
doronshadmi said:
enough dots?
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No, not enough dots.

We are talking about an endless (edgeless) straight line, which is an atom (it is not composed by dots).
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We? No, we were doing no such thing. You might have thought you were, but we were not. Moreover, your "endless (edgeless) straight line" remains an undefined concept.
 
jsfisher said:
No, not enough dots.



We? No, we were doing no such thing. You might have thought you were, but we were not. Moreover, your "endless (edgeless) straight line" remains an undefined concept.
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Because definition, according to your reasoning, is limited to a pile of dots.
 
jsfisher said:
Yes, those would be excellent samples of gibberish you posted.
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jsfisher, in order to fail to respond, you first have to respond.

Since The Man did not respond at all, he did not fail to respond, as you claim.

As for your "reply", it is not a reply unless you provide a detailed respons.
 
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doronshadmi said:
jsfisher, in order to fail to respond, you first have to respond.

Since he did not respond at all, he did not fail to respond, as you claim.

As for your "reply", it is not a reply unless you provide a detailed respons.
Click to expand...

You really, really need that English dictionary. You failed to understand the phrase, "fail to respond."
 
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