Deeper than primes

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jsfisher said:
How will that assist you in understanding simple English statements?
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It does not limited to English.

In order to fail you first have to try to response.

The one who does not try, can't succeed or fail.

Simple common sense.
 
doronshadmi said:
If you cut a line, you get a complex, which is a dot AND a line
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Well,the words are English,but that makes no sense at all.I've just looked up the definition of the word complex in the Greater Oxford Dictionary and no mention was made of your private definition.I suppose that the compilers were fools too,in your opinion.It must feel wonderful to be smarter than Cantor,Newton,Hilbert and Pythagoras all rolled together.
 
If I was an architect designing a complex and I got a complex about it and then complications set in,would I have a complex complex complex?
 
doronshadmi said:
Simple common sense.
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Ya gotta love common sense.

A conversation with my ex-girlfriend:

Apathia: Ok, there's this hotel, but it's special. It has an infinite number of rooms. And today all the rooms are taken.

Gale: Stop right there! This hotel you are talking about is a single building. It's only one thing. So it can't be infinite!

Apathia: Of course it's only one hotel, but it contains an infinite amount of rooms.

Gale: How the heck could it do that. Infinity means there are always more. There are always more rooms that aren't in the hotel.

Apathia: But this hotel already includes all the rooms in infinity.

Gale: Your head is full of stupid abstractions. An infinity can never fit in a finite space.

Apathia OK, how about this ruler? How many points are there between 0 and 1 inch?"

Gale: None.

Apathia: Include fractions.

Gale: As many as you make. But get this: you never make an infinite number of fractions.

Apathia: Mathematically speaking, Gale. How many points are there?

Gale: I just told you.

Apathia: Try this then: an arrow is shot at a target. It must pass each point before it strikes the bull's eye.

Gale: Not that old thing! The arrow isn't stopping to mark any points. It's not dividing the space into fractions. It's just flying the finite space between the bow and the target.

As I've always told you, Rei, you think too much.
That Doron guy has it right.

Apathia (once known to Gale as "Rei"): So you agree with Doron that a decimal point followed by an infinite number of 9s never amounts to one?
[Rei shows her the typical grade school proof]

Gale: It's just a convenience. Like Pi is just 3.14. A shortcut that's all. It never really equals one.

The thing you're not getting, Rei is that infinity is just an idea in your head. Another confused idea!

Apathia You've presented the concept of Potential Infinity much the way Aristotle would. Doron though presents an Actual, or in his case, an Absolute Infinity.

Gale: Well I think all this infinity talk is just an intellectual fantasy!
His head is full of stuff, just like yours.

BTW, I spoke to the fairies about our relationship, and we need to have a talk.

Happy New Year to all!
May your heads be full of infinities, not fairies. :wackylaugh:
 
doronshadmi said:
In that case please show us your reply to http://www.internationalskeptics.com/forums/showpost.php?p=5400999&postcount=7409 and

http://www.internationalskeptics.com/forums/showpost.php?p=5402429&postcount=7429
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I never claimed to have replied directly to those posts, but I certainly didn’t miss them. The errors inherent in those posts have been brought up multiple times prior to those posts, that you simply choose to ignore them and expect a reply is simply your problem. I will leave it to you to review this thread and find where those issues were specifically addressed by me and/or others. However if you still insist on some response from me, the one you quoted about sums it up for me.

The Man said:
You think inccorectly
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By the way, have you figured out what is infinite about your “endless atomic straight line” yet?
 
doronshadmi said:
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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I thought that you would make a case for the braces being as much as qualified to reflect upon the size of sets as the parentheses are. Actually there is a "proof" of that, but it is a bit difficult to construe. Maybe if I toss it your way, you can find a flaw in it, when you took on Georg Cantor.

First comes kind of fuzzy comparison that needs a bit of explaining.

PARENTHESES are to () = 5 as BRACES are to {} = ?

The term () = 5 means that the parentheses will widen to accommodate some 5 elements, thus creating set P with size P = 5. It figures that there must be some reason why () = 5 and not () = 6 or any other option. But since the word "parentheses" is the first part of the premise, the choice () = 5 must depend on the meaning of the word or perhaps even on the shape of "()". Without knowing the logic of the premise, the conclusion regarding the braces cannot be reached. So what kind of meaningful relation does exist between parenthesis() and 5?

Beat me. I don't know. If there is some relationship, then it surely doesn't stick out.

Our educational system doesn't allow much room for going ahead to speculate further upon the subject: either you know or you don't; either you pass or fail the exam. But there is other environment where the comparison can be transferred to. That's the one where questions such as if you can't do this, is there anything else you can do? can be asked.

Well, I can solve the comparison on my own terms; I will simply establish the kind of logic that ties premise A is to B and solve the conclusion.

PARENTHESES are to () = 11 as BRACES are to {} = 6

For the above comparison to be true, I simply fill the empty set () with the word "parenthesis," so () = 11 would hold true.

PARENTHESIES are to (PARENTHESES) = 11 as BRACES are to {BRACES} = 6

In other words, set P has size 11, or P = 11, coz there are 11 elements/letters in it, and the same relation is applied in the conclusion where set B = 6.

Solving something that you actually invent is very easy, isn't it? But this option to the original comparison where () = 5, makes its solution apparent.

Apparent? It may provide some clue, but it's not apparent. Why would it be? But the kind of illogical usage of the word "apparent" in the context is actually the most visible clue to the solution where we compare things. And so, let's compare . . .

APPAREnt <=> APPAREl

The size, or cardinality, of sets relates well to apparel, which people buy to according to the size to fit them. And so we can pull the trigger.

PARENTHESES are to () = 5 as BRACES are to {} = 3

The justification lies in "some atoms becoming non-local."

--RE--HE-ES are to (PANTS) = 5 as ---CES are to {BRA} = 3

And that concludes the proof that both braces and parentheses are equally well-suited for holding elements of sets that have different sizes.
 
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dafydd said:
Well,the words are English,but that makes no sense at all.I've just looked up the definition of the word complex in the Greater Oxford Dictionary and no mention was made of your private definition.I suppose that the compilers were fools too,in your opinion.It must feel wonderful to be smarter than Cantor,Newton,Hilbert and Pythagoras all rolled together.
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Again you take the personal view of the considered subject.

OM’s notion is very simple:

0 is actual-finite (it is the minimal aspect of the atomic state)

∞ is actual-infinite (it is the maximal aspect of the atomic state)

∞\0 is a complex (it is the result of the linkage of the extreme actual aspects of the atomic state).

Cantor’s framework is closed under ∞\0 , where ∞\0 is used both as premise and conclusion.

As a result we get a framework that is based on circular reasoning.
Since OM is at least 0 < ∞\0 < ∞, it is not closed under the mentioned circular reasoning.

By 0 < ∞\0 < ∞ we get an open system, which is not a deductive-only framework, and it is a paradigm-shift of (∞\0)-only framework.

You will not find 0 < ∞\0 < ∞ in any currently known knowledge-base, exactly because it is not the agreed (∞\0)-only framework.
 
jsfisher said:
My point was that "a" doesn't appear as a member of the power set. It may be a member of some of the members of the power set, but that isn't the same thing.
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jsfisher's framework is a flat one, only the first level of a given collection is considered.
 
epix said:
I thought that you would make a case for the braces being as much as qualified to reflect upon the size of sets as the parentheses are. Actually there is a "proof" of that, but it is a bit difficult to construe. Maybe if I toss it your way, you can find a flaw in it, when you took on Georg Cantor.

First comes kind of fuzzy comparison that needs a bit of explaining.

PARENTHESES are to () = 5 as BRACES are to {} = ?

The term () = 5 means that the parentheses will widen to accommodate some 5 elements, thus creating set P with size P = 5. It figures that there must be some reason why () = 5 and not () = 6 or any other option. But since the word "parentheses" is the first part of the premise, the choice () = 5 must depend on the meaning of the word or perhaps even on the shape of "()". Without knowing the logic of the premise, the conclusion regarding the braces cannot be reached. So what kind of meaningful relation does exist between parenthesis() and 5?

Beat me. I don't know. If there is some relationship, then it surely doesn't stick out.

Our educational system doesn't allow much room for going ahead to speculate further upon the subject: either you know or you don't; either you pass or fail the exam. But there is other environment where the comparison can be transferred to. That's the one where questions such as if you can't do this, is there anything else you can do? can be asked.

Well, I can solve the comparison on my own terms; I will simply establish the kind of logic that ties premise A is to B and solve the conclusion.

PARENTHESES are to () = 11 as BRACES are to {} = 6

For the above comparison to be true, I simply fill the empty set () with the word "parenthesis," so () = 11 would hold true.

PARENTHESIES are to (PARENTHESES) = 11 as BRACES are to {BRACES} = 6

In other words, set P has size 11, or P = 11, coz there are 11 elements/letters in it, and the same relation is applied in the conclusion where set B = 6.

Solving something that you actually invent is very easy, isn't it? But this option to the original comparison where () = 5, makes its solution apparent.

Apparent? It may provide some clue, but it's not apparent. Why would it be? But the kind of illogical usage of the word "apparent" in the context is actually the most visible clue to the solution where we compare things. And so, let's compare . . .

APPAREnt <=> APPAREl

The size, or cardinality, of sets relates well to apparel, which people buy to according to the size to fit them. And so we can pull the trigger.

PARENTHESES are to () = 5 as BRACES are to {} = 3

The justification lies in "some atoms becoming non-local."

--RE--HE-ES are to (PANTS) = 5 as ---CES are to {BRA} = 3

And that concludes the proof that both braces and parentheses are equally well-suited for holding elements of sets that have different sizes.
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Still you are closed under the concept of Collection.

______ is not a collection but it is the non-local atomic aspect that cannot be captured by any given domain ( generally notated as "(" and ")" ).

By using geometrical representation of this notion:

. < ___ \ . < ___ actually = 0 < ∞\0 < ∞
 
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The Man said:
I never claimed to have replied directly to those posts, but I certainly didn’t miss them. The errors inherent in those posts have been brought up multiple times prior to those posts, that you simply choose to ignore them and expect a reply is simply your problem. I will leave it to you to review this thread and find where those issues were specifically addressed by me and/or others. However if you still insist on some response from me, the one you quoted about sums it up for me.
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You did exactly nothing about OM, because your reasoning is closed under Complexity, without the needed understanding of what enables Complexity, in the first place.

The Man said:
By the way, have you figured out what is infinite about your “endless atomic straight line” yet?
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Given any domain, an endless atomic straight line can’t be captured by it.
 
Apathia said:
Apathia You've presented the concept of Potential Infinity much the way Aristotle would. Doron though presents an Actual, or in his case, an Absolute Infinity.
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Not correct.

OM is 0 < ∞\0 < ∞, where 0 represents actual-finite, ∞\0 represents Complexity and ∞ represents actual-infinite.
 
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jsfisher said:
You'd need to be very specific about what contradictory statements you think I made, first.
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From one hand you claim that mathematical models are deductive context-dependent frameworks, and on the other hand you claim that deductive context-dependent frameworks can contradict each other as if they belong to a one framework.

So "disjoint AND joint framework" is a contradiction.
 
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The term “all” is relevant only if some mapping is available more than once w.r.t some given non-empty collection.

This is not the case with an infinite collection.

In other words, "all" is a conclusion and not a premise in the case of non-empty collections.

Cantor used “all” as a premise in order to define its trasfinite system. As a result he forced completeness on a naturally incomplete mathematical concept like infinite collection.
 
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doronshadmi said:
You did exactly nothing about OM, because your reasoning is closed under Complexity, without the needed understanding of what enables Complexity, in the first place.
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Ah, another Doron contradictory evasion tactic, labeling “your reasoning is closed under Complexity” and claiming one must simply accept your notions of “understanding of what enables Complexity, in the first place” in order to show your notions as both self-inconsistent and generally inconsistent. Doron you do exactly nothing about “OM” except to simply refute it yourself, as evidenced by your subsequent assertion.

doronshadmi said:
Given any domain, an endless atomic straight line can’t be captured by it.
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Well that still doesn’t say what is infinite about your “endless atomic straight line”. In fact it says that nothing is actually infinite about your “endless atomic straight line” since it “can’t be captured by” even an infinite domain (being “any domain”)”. Further it stipulates that your “endless atomic straight line” “can’t be captured by” itself as the domain (also being “any domain”). There goes your X = X “sameness reasoning” (more specifically circular reasoning) basis at least for your “endless atomic straight line”. Just your usual self-contradictory nonsensical gibberish Doron, the only thing you actually do with your OM is directly contradict your OM.


That’s the real rub of it Doron, you accept your own notions, but you are the most consistent direct opponent of your own notions. No one need conform to your “understanding of what enables Complexity, in the first place” when you claim to yourself yet simply can not demonstrate that you actually do and continue to directly oppose your own assertions of your “understanding of what enables Complexity, in the first place” with your “understanding of what enables Complexity, in the first place”.
 
doronshadmi said:
Not correct.

OM is 0 < ∞\0 < ∞, where 0 represents actual-finite, ∞\0 represents Complexity and ∞ represents actual-infinite.
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Once again, as pointed out to you before, Zero is not always considered finite.


http://dictionary.reference.com/browse/finite

fi⋅nite
–adjective

2. Mathematics.
a. (of a set of elements) capable of being completely counted.
b. not infinite or infinitesimal.
c. not zero.
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http://en.wikipedia.org/wiki/Finite
Finite
From Wikipedia, the free encyclopedia
Finite is the opposite of infinite. It may refer to:
• Having a finite number of elements: finite set
• Being a finite number, so not equal to ±∞; for example, all real numbers are finite
• In a stronger sense, being a value that is neither infinite nor infinitesimal or zero; in this sense all real numbers except 0 are finite
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If you include 0 as finite then all real numbers (your complex) are equally as finite, in fact all other real numbers (your complex) are finite, but 0 need not be considered finite.


So what exactly is finite about 0 in your OM? How does your “actual-finite” differ from any finite real number?

What about -∞? Should not your OM be -∞ < -∞\0 < 0 < ∞\0 < ∞? Perhaps that is just too complex for you? The domain of [0, ∞] would represent a ray not a line, your “endless atomic straight line” is actually represented as the domain [-∞,∞].

Actual math is more consistent, useful and efficient than your OM as your entire “OM is 0 < ∞\0 < ∞” (as asserted by you) is contained in the domain of [-∞,∞].
 
doronshadmi said:
Please stop talking to yourself.
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Ah! I see you are applying the Pee Wee Herman Gambit. How's that working for you?

Seriously, though, stop blaming others just because you don't understand the meaning of something (be it Mathematics or English). You blew it with your aside on the meaning of "fail to respond". Move on without further self-imposed embarrassment.
 
doronshadmi said:
From one hand you claim that mathematical models are deductive context-dependent frameworks, and on the other hand you claim that deductive context-dependent frameworks can contradict each other as if they belong to a one framework.

So "disjoint AND joint framework" is a contradiction.
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Where did I make such statements? Please point out any post by me where I even used the phrase, "deductive context-dependent frameworks", let along any of the other claims you attribute to me.
 
doronshadmi said:
Not correct.

OM is 0 < ∞\0 < ∞, where 0 represents actual-finite, ∞\0 represents Complexity and ∞ represents actual-infinite.
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Doron, your "actual-infinite" is, as you have repeated elsewhere, an "atom," a fundamental, "ontological," non-composite, independent, "entity."
It is indivisible and cannot be contained.

This is why it's an Absolute, as opposed to a potential.

I see why you don't want to use the term, Absolute. You want to maintain the either-or of "actual" as opposed to "potential." The only actual, real Infinite, in your book, is the ∞-Atom.
And That, though it isn't dependent upon the finite, is only present when it is working in pair with the 0-Atom, or, as I called it, the Absolute Finite.

Since you insist that the ∞-Atom and the 0-Atom are mutually independent and ultimate building blocks, they are your absolutes. The term applies.

It's fine with me if you want to assert that the only real and Actual Infinity is what philosophers call Absolute Infinity. And that "completed" infinities and Cantor's Transfinites have no actuality.
But you will continue to fail to communicate your views as long as you can't see how others use and define the words you impose your own divergent meanings upon.
As you've said so many times, you have to get beyond the words.

My xgf confused the concepts of potential, absolute, and actual infinity
She starts out thinking its the Absolute and then asserts it as a conceptual fiction, just the same path Aristotle took in coming to the conclusion that there was no Actual or Absolute Infinity in reality, just a potentiality in thought.

But you do assert an actuality for Infinity, actuality for Absolute Infinity (not for Cantor's "completed" infinities). Modern Mathematics treats Cantor's transfinites as actuals, rather than potentials.
Your Actual Infinity is Absolute, "ontological," and metaphysical (all words for the same pov.)

The unique thing that you introduce is the cooperation of The Absolute Infinite with the Absolute Finite.
These as a participant pair is your brand new contribution to the subject of Infinity.
Presenting Absolute Infinity as the only actual infinity is not new, but The Finite as an Absolute is unique.
And the pairing of them to produce number (Organic Numbers); that's your baby.

And it comes to some radically new ways of thinking about reality.
For example, The Actual Finite, as opposed to what is just potentially finite and never actually is.
All things (as "researchable" objects) are "complexes" that cannot be said to be actually finite any more than they can be said to be actually infinite.
 
The Man said:
Once again, as pointed out to you before, Zero is not always considered finite.


http://dictionary.reference.com/browse/finite






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



If you include 0 as finite then all real numbers (your complex) are equally as finite, in fact all other real numbers (your complex) are finite, but 0 need not be considered finite.


So what exactly is finite about 0 in your OM? How does your “actual-finite” differ from any finite real number?

What about -∞? Should not your OM be -∞ < -∞\0 < 0 < ∞\0 < ∞? Perhaps that is just too complex for you? The domain of [0, ∞] would represent a ray not a line, your “endless atomic straight line” is actually represented as the domain [-∞,∞].

Actual math is more consistent, useful and efficient than your OM as your entire “OM is 0 < ∞\0 < ∞” (as asserted by you) is contained in the domain of [-∞,∞].
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Neat. 0 isn't necessarily finite.
Doron might agree that 0 is as a number is not Actually Finite.

Again we have a very disjoint way of using words going on in this entire thread. 0 and ∞ are for all the mathematicians posting here, mathematical objects.
Doron uses them as symbols or metaphors for "ontological" concepts.
But he also uses them as mathematical objects. And the frequent juxtaposition of two different modes of meaning and their separate linguistic significance creates confusion and contradiction.
 
Apathia said:
Doron, your "actual-infinite" is, as you have repeated elsewhere, an "atom," a fundamental, "ontological," non-composite, independent, "entity."
It is indivisible and cannot be contained.

This is why it's an Absolute, as opposed to a potential.

I see why you don't want to use the term, Absolute. You want to maintain the either-or of "actual" as opposed to "potential." The only actual, real Infinite, in your book, is the ∞-Atom.
And That, though it isn't dependent upon the finite, is only present when it is working in pair with the 0-Atom, or, as I called it, the Absolute Finite.

Since you insist that the ∞-Atom and the 0-Atom are mutually independent and ultimate building blocks, they are your absolutes. The term applies.

It's fine with me if you want to assert that the only real and Actual Infinity is what philosophers call Absolute Infinity. And that "completed" infinities and Cantor's Transfinites have no actuality.
But you will continue to fail to communicate your views as long as you can't see how others use and define the words you impose your own divergent meanings upon.
As you've said so many times, you have to get beyond the words.

My xgf confused the concepts of potential, absolute, and actual infinity
She starts out thinking its the Absolute and then asserts it as a conceptual fiction, just the same path Aristotle took in coming to the conclusion that there was no Actual or Absolute Infinity in reality, just a potentiality in thought.

But you do assert an actuality for Infinity, actuality for Absolute Infinity (not for Cantor's "completed" infinities). Modern Mathematics treats Cantor's transfinites as actuals, rather than potentials.
Your Actual Infinity is Absolute, "ontological," and metaphysical (all words for the same pov.)

The unique thing that you introduce is the cooperation of The Absolute Infinite with the Absolute Finite.
These as a participant pair is your brand new contribution to the subject of Infinity.
Presenting Absolute Infinity as the only actual infinity is not new, but The Finite as an Absolute is unique.
And the pairing of them to produce number (Organic Numbers); that's your baby.

And it comes to some radically new ways of thinking about reality.
For example, The Actual Finite, as opposed to what is just potentially finite and never actually is.
All things (as "researchable" objects) are "complexes" that cannot be said to be actually finite any more than they can be said to be actually infinite.
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It is really nothing new Apathia, the only thing Doron actually brings to the table is his insistence on inconsistency, which isn’t new either. If you recall on the other thread I mentioned the infinite/finite relation of a line segment and the finite/ infinite relation of a line (or ray). A segment is finite in length yet infinite in locations it encompasses. That finite length is in fact based on a finite number of locations (2) that are the boundaries of that segment. A line or ray is an infinite sum of such finite segments. All very consistent, simple, useful and standard, all founded on the concept of a point, a finite number of points and an infinite number of points. All Doron brings to it is his misunderstanding and inconsistency.
 
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Apathia said:
Neat. 0 isn't necessarily finite.
Doron might agree that 0 is as a number is not Actually Finite.

Again we have a very disjoint way of using words going on in this entire thread. 0 and ∞ are for all the mathematicians posting here, mathematical objects.
Doron uses them as symbols or metaphors for "ontological" concepts.
But he also uses them as mathematical objects. And the frequent juxtaposition of two different modes of meaning and their separate linguistic significance creates confusion and contradiction.
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No Apathia, Doron simply has a disjoint way of using words. Instead of taking the time and actually finding the words to say what he means he simply redefines terms and concepts even to the point of where they lose any specific meaning. Quite frankly I couldn’t give a flying handshake about Doron’s “metaphors for "ontological" concepts” or if he chooses to be self-consistent and/or generally consistent with them, and if that were the extent of his assertions I most likely would not be here to debate those issues with him. However, Doron makes very specific claims about math and mathematical objects that simply are not true and as you note ‘juxtaposes the two’ (I would be more inclined to say deliberately confuses the two), but that is his problem. Unfortunately, regardless of the significance, consistency, or lack of either, one may consider for Doron’s “metaphors for "ontological" concepts”, we would be remiss and complacent if we simply allowed his misrepresentations and misunderstanding (deliberate or otherwise) about math, mathematical objects and concepts to continue unabated.
 
The Man said:
No Apathia, Doron simply has a disjoint way of using words. Instead of taking the time and actually finding the words to say what he means he simply redefines terms and concepts even to the point of where they lose any specific meaning. Quite frankly I couldn’t give a flying handshake about Doron’s “metaphors for "ontological" concepts” or if he chooses to be self-consistent and/or generally consistent with them, and if that were the extent of his assertions I most likely would not be here to debate those issues with him. However, Doron makes very specific claims about math and mathematical objects that simply are not true and as you note ‘juxtaposes the two’ (I would be more inclined to say deliberately confuses the two), but that is his problem. Unfortunately, regardless of the significance, consistency, or lack of either, one may consider for Doron’s “metaphors for "ontological" concepts”, we would be remiss and complacent if we simply allowed his misrepresentations and misunderstanding (deliberate or otherwise) about math, mathematical objects and concepts to continue unabated.
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:wackyembarrassed:
You've got me there.
Actually it's Doron who's going to throw that very pie in my face if and when he replies to my post.
 
Apathia said:
:wackyembarrassed:
You've got me there.
Actually it's Doron who's going to throw that very pie in my face if and when he replies to my post.
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One of the ontological classics of slapstick comedy, the pie in the face. I'm sure Doron can make a dichotomy and complex out of that too. The pie singular, unchanging on its own and unique. The face infinite in variability and the pie/face combination the actual researchable aspect of comedy.
 
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doronshadmi said:
Still you are closed under the concept of Collection.

______ is not a collection but it is the non-local atomic aspect that cannot be captured by any given domain ( generally notated as "(" and ")" ).

By using geometrical representation of this notion:

. < ___ \ . < ___ actually = 0 < ∞\0 < ∞
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Aha. Well, I kind of suspected that you would find some inconsistence ( "Still you are closed under the concept of Collection.") in my proof concerning the usage of parentheses and braces as a home to the elements of sets. We may look at some symbolism for centuries but still may not be aware of flawed logic in it. For example, words are often symbolized by their initials, such as 'O' for "open" and 'C' for the opposite case "close." But 'O' is hardly an opened figure, as opposed to 'C', which very much resembles a semi-circle and, unlike 'O', can be exited from within. So that's a paradox that concerns opposites, which indirectly leads toward the usage of parentheses and braces under different conditions.

There is a big difference between 0 and 1 as opposed to a difference between 1 and 2, coz the former case reflects upon the difference between absence and presence, whereas the latter case is just a difference between a number of items that are already present. So if we consider two sets; one which is empty and the other, which is not, the non-empty sets should be symbolized by a set having just one element: empty set = {} and non-empty set = {a}.

Why would I use braces rather than parentheses for a demonstration?

I think braces are better suited for holding one element, coz the choice of the element is determinable. There is no problem for parentheses being used for an empty set,

PARENTHESES: ()

but difficulties arise when () is supposed to be filled with one element -- the entropy is much higher than in the usage of braces.

BRACES: {}
BRA-ES: {C}

Since 'C' divides the word "braces" into two parts -- "bra" and "es"-- it is logically chosen to fill the empty set with respect to the legible part "bra" in conjunction with the shape 'C' symbolizing a circular action with the intention to complete it into the state 'O'.
http://www.insidesocal.com/bargain/BraFitting.jpg

Of course, it can go the other way -- the way I prefer: from 'O' to 'C'. But the theory of sets wasn't invented to make someone smile, and so I leave the question of button/unbutton to your formal treatment.
 
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The Man said:
Ah, another Doron contradictory evasion tactic, labeling “your reasoning is closed under Complexity” and claiming one must simply accept your notions of “understanding of what enables Complexity, in the first place” in order to show your notions as both self-inconsistent and generally inconsistent. Doron you do exactly nothing about “OM” except to simply refute it yourself, as evidenced by your subsequent assertion.



Well that still doesn’t say what is infinite about your “endless atomic straight line”. In fact it says that nothing is actually infinite about your “endless atomic straight line” since it “can’t be captured by” even an infinite domain (being “any domain”)”. Further it stipulates that your “endless atomic straight line” “can’t be captured by” itself as the domain (also being “any domain”). There goes your X = X “sameness reasoning” (more specifically circular reasoning) basis at least for your “endless atomic straight line”. Just your usual self-contradictory nonsensical gibberish Doron, the only thing you actually do with your OM is directly contradict your OM.


That’s the real rub of it Doron, you accept your own notions, but you are the most consistent direct opponent of your own notions. No one need conform to your “understanding of what enables Complexity, in the first place” when you claim to yourself yet simply can not demonstrate that you actually do and continue to directly oppose your own assertions of your “understanding of what enables Complexity, in the first place” with your “understanding of what enables Complexity, in the first place”.
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A domain is based on limits.

Since an endless atomic straight line is limitless, it can't be capured by any given domain.

A point, for example is the most limited domain.

An endless atomic straight line is the minimal limitless form.

In other words The Man, your reasoning can't get it.
 
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Apathia said:
Neat. 0 isn't necessarily finite.
Doron might agree that 0 is as a number is not Actually Finite.

Again we have a very disjoint way of using words going on in this entire thread. 0 and ∞ are for all the mathematicians posting here, mathematical objects.
Doron uses them as symbols or metaphors for "ontological" concepts.
But he also uses them as mathematical objects. And the frequent juxtaposition of two different modes of meaning and their separate linguistic significance creates confusion and contradiction.
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0 is a number only under Relation\Elelemnt Interaction where REI is a complex state.

The left 0 in the expression "0 < ∞\0 < ∞" is not a number but it is actual finite, exactly as the right ∞ of this expression is not a number but it is actual infinite.
 
The Man said:
Once again, as pointed out to you before, Zero is not always considered finite.


http://dictionary.reference.com/browse/finite






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



If you include 0 as finite then all real numbers (your complex) are equally as finite, in fact all other real numbers (your complex) are finite, but 0 need not be considered finite.


So what exactly is finite about 0 in your OM? How does your “actual-finite” differ from any finite real number?

What about -∞? Should not your OM be -∞ < -∞\0 < 0 < ∞\0 < ∞? Perhaps that is just too complex for you? The domain of [0, ∞] would represent a ray not a line, your “endless atomic straight line” is actually represented as the domain [-∞,∞].

Actual math is more consistent, useful and efficient than your OM as your entire “OM is 0 < ∞\0 < ∞” (as asserted by you) is contained in the domain of [-∞,∞].
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The Man,

0 is the minimal Cardinality (actual finite) as ∞ is the maximal Cardinality (actual infinite).

You do not get 0 < ∞\0 < ∞ in terms of Cardinality.
 
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Apathia said:
It's fine with me if you want to assert that the only real and Actual Infinity is what philosophers call Absolute Infinity. And that "completed" infinities and Cantor's Transfinites have no actuality.
But you will continue to fail to communicate your views as long as you can't see how others use and define the words you impose your own divergent meanings upon.
As you've said so many times, you have to get beyond the words.
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Let us avoid “that is beyond words” because the posters here are closed under their thoughts.

Definition A:

A non-empty collection is considered as complete if each element of it is both the beginning and the end of this collection.

For example a collection of beads along a closed string is complete if each bead along the closed string is both the beginning and the end of this collection of beads.

Now please replace beads by points and a closed string by a closed curve.

A collection of points along a closed curve is complete if each point along the closed curve is both the beginning and the end of this collection of points.

Only a finite collection has this property.

Since no point of an infinite collection of points along a closed curve is both the beginning and the end of this collection, then this collection is incomplete by definition A.
 
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epix said:
Aha. Well, I kind of suspected that you would find some inconsistence ( "Still you are closed under the concept of Collection.") in my proof concerning the usage of parentheses and braces as a home to the elements of sets. We may look at some symbolism for centuries but still may not be aware of flawed logic in it. For example, words are often symbolized by their initials, such as 'O' for "open" and 'C' for the opposite case "close." But 'O' is hardly an opened figure, as opposed to 'C', which very much resembles a semi-circle and, unlike 'O', can be exited from within. So that's a paradox that concerns opposites, which indirectly leads toward the usage of parentheses and braces under different conditions.

There is a big difference between 0 and 1 as opposed to a difference between 1 and 2, coz the former case reflects upon the difference between absence and presence, whereas the latter case is just a difference between a number of items that are already present. So if we consider two sets; one which is empty and the other, which is not, the non-empty sets should be symbolized by a set having just one element: empty set = {} and non-empty set = {a}.

Why would I use braces rather than parentheses for a demonstration?

I think braces are better suited for holding one element, coz the choice of the element is determinable. There is no problem for parentheses being used for an empty set,

PARENTHESES: ()

but difficulties arise when () is supposed to be filled with one element -- the entropy is much higher than in the usage of braces.

BRACES: {}
BRA-ES: {C}

Since 'C' divides the word "braces" into two parts -- "bra" and "es"-- it is logically chosen to fill the empty set with respect to the legible part "bra" in conjunction with the shape 'C' symbolizing a circular action with the intention to complete it into the state 'O'.
http://www.insidesocal.com/bargain/BraFitting.jpg

Of course, it can go the other way -- the way I prefer: from 'O' to 'C'. But the theory of sets wasn't invented to make someone smile, and so I leave the question of button/unbutton to your formal treatment.
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epix,

I am not talking about the shapes of the symbols.

I am talking about the notions that stand at their basis.

A domain is based on the notion of limitation, no matter what shape is used to represent it.

Actual infinity is the actual domainless and actual finite is the actual domain.

The minimal representation of these notions is “.” for actual domain and “…_____...” for actual domainless.

This is simple and straightforward, but any other representation can be used as well.
 
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doronshadmi said:
A domain is based on limits.

Since an endless atomic straight line is limitless, it can't be capured by any given domain.

A point, for example is the most limited domain.

An endless atomic straight line is the minimal limitless form.

In other words The Man, your reasoning can't get it.
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How is the limitation to one dimension “the minimal limitless form”?

The fact still remains Doron that “your reasoning can't get it”.

If you simply want to ignore the limitations of a singular dimensional domain, like a line, that is your prerogative, but it does not make it any less limited to that singular dimension or any less of a domain.
 
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