A collection of infinitely many distinct elements

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Apathia said:
Not at all. Doron wants to do mathematics, not philosophy. He wants collaborating mathematicians who will help him devolop his basic approach into a functioning system. (or so he has said in this and other forums.)
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If he wants to do mathematics, he should at least show a high-school level of mathematics knowledge. He doesn't. He only spouts wooish ********, in bad English to boot. If he wants real mathematicians to help, he should start with not antagonizing them and have proper respect for actual, real mathematics. Everybody has said so numerous times, but he keeps on antagonizing, and he refuses to answer questions. That's quite clear by now.

Apathia said:
But I'm not being helpfull. (Not that I could, even if I were a mathematician.)
Because I got an interesting story idea from all this, I'll still be following along. But I think I should draw back to lurking as a was for while.
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Fine. Then the thread really only consists of endless bickering. Time to have it off to AAH.
 
Apathia said:
Not at all. Doron wants to do mathematics, not philosophy.
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No he doesn't.

He wants collaborating mathematicians who will help him devolop his basic approach into a functioning system. (or so he has said in this and other forums.)
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Right. He doesn't want to do mathematics. He doesn't want to learn anything. He doesn't want to be bothered with any actual effort. He just wants to be praised for his mystical insights.

He's come to the wrong forum for that.
 
doronshadmi said:
You have two main problems:

1) You can understand something only if it is represented by words.

2) You can understand the words only by using a step-by-step thinking.

Since the organic paradigm is a parallel\serial thinking you are unable to get it by your (1),(2) point od view.
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Please explain the following picture without using words. I have removed the wording inside to make it easier for you.


 
Apathia said:
Life here might be a little easier if that was the case. Then there would be a source of definitions everybody use to disciper Doron.
Doron his picked up various terms from different sources over his career, but none of them mean what they did in their source. He's also generated his own vocabulary which is ever changing. .
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I agree completely, although being more of a student of philosophy then the others of us on this science and math sub forum, your certainly seem to have a better grasp of that changing terminology and again I thank you for the explanations you have given.


Apathia said:
It's more about getting the method of thinking in terms of complementary polar opposites. For example, take "interior" and pair it with its opposite "exterior" such that they are independent concepts with discrete combinations making a set of circumstances (Remember Doron's stick and the circle analogy in this thread?) Or make it subject and object, or observer and obserrved.
Take what ever pair works the best for you to see the method that Doron says is universal to cognition. He says this is the way we think, by manipulating polar opposite terms. .
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But that is just my point that polar opposite terms are just that opposites. North of the equator you are north, south of the equator you are south, as the equator is just a theoretical line so you can never be both north and south. Yes this is a physical application of theoretical ascriptions, but that is exactly what doronshadmi is trying to do with his “wavicle”. If you want complimentary opposites then it is more like yin and yang or the geometric examples of infinite and finite that I gave before. They are not polar opposites but complimentary opposites that stem from each other. That doronshadmi is now attempting to combine the two, polar (or absolute) opposites and complimentary (or intermixed opposites) is an absolutistic escapism philosophy. Why worry about paradoxes or contradictions, the very aspects that any philosophy should specifically and in detail deal with. Just cut yourself a trap door out the bottom combining the specifically absolute with the specifically not absolute and escape dealing with those issue. Unfortunately, this just gets you out of the frying pan and into the fire.
 
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doronshadmi said:
Fullness is too strong for finding a place to put something because it has no room for more things.

You also cannot find any place at total emptiness, because place is stronger than emptiness.

"A place", "looking for", "to put something", "perspective", "locality", "non-locality" , etc... can be found as meaningful concepts only if totality is avoided.
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This does not help you, but only confirms the utter subjective nature of your assertions and definitions.

Why is "place" stronger then "emptiness", when it might in fact be both, a totally empty place? Just so it fits with your subjective assertions and definitions? You are the only one who seems to be avoiding the contradictions inherent in your subjective assertions and definitions.
 
doronshadmi said:
Actual Mathematics is the art of the abstraction.

Since you cannot get the notion of line\point logical connective\proposition abstraction as non-locality\locality, you are not doin' actual mathematics.
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Little 10 Toes said:
Please explain why 1+1=2 is need of any "line\point logical connective\proposition abstraction as non-locality\locality"?
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Actually, Mathematics is just a language (as I have said before), and like all languages it is dependent on the definitions (not abstractions) inherent in that language. Like all languages definitions are really just common usage, and although this does add an apparent air of abstraction, the art, like any language, is not in the application of the abstractions but in the application of the definitions (or usage common and uncommon). The more abstract your usage is the less others will understand you, but again just a matter of perspective, as an uncommon usage to you might be common to someone else.

For example 2+2 = 5 would normally be considered incorrect in an absolutist mathematical sense. As a language we must consider what it is that each of these symbols might represent. In the language of English, if it is two apples plus two apples equals five apples then it is certainly incorrect. However, if it is two one dollar apples plus two one and a half dollar apples equals five dollars spent on apples, then it is correct (but expensive apples). Definitions (or usage common and uncommon) even when applied to abstractions make or break what you say with any language and any philosophical notions.
 
doronshadmi said:
Mathematics and Physics are both researchable frameworks, based on atom's self reference.

The atom's self state (which is not atom's self reference) is not researchable, but I am talking about the researchable frameworks, where Math & science deals in "facts" and theories that depend on the researcher's self awareness.
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As you've now brought physics into the mix, and therefore tied this to phenomena, you must be using 'atom' in the usual sense any physicist would understand that noun. Namely the thing made of electrons, neutrons and protons. Any other meaning of the noun 'atom' in this context would be insane.

So, please clarify what an atom's self-reference is.
 
jsfisher said:
How so? Cantor's proof relies on exactly one set D. You misunderstand the method and claim Cantor's proof uses infinitely many such sets.
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If there is one set called D, then Cantor's proof definitely has nothing to do with X < P(X) where both of them are sets with non-finite members.

You do not get that x of 0 =< x =< 1 is equivalent to D of {{}, … , {a, b, c, d, …}}, and each D (which is a member of P(X) is constructed in such a way that is not no map with any of X members.

This result can be generalized to P(X) < P(P(X)), to P(P(X)) < P(P(P(X))) , etc. … ad infinitum.

Furthermore, P(X) members are actually constructed by this beautiful and simple

[latex]$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$[/latex]​

which is not a single outcome, as you claim, but it is a general method to define P(X) out of X.

jsfisher said:
Ok, so what? 2, 3 and 4 are each members of N, but that doesn't define the set N. Your D sets do not define P(X) either.
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Let alone show your comparison between N (which is not closed from above, for example 1,2,3,4,…) http://www.internationalskeptics.com/forums/showpost.php?p=3654898&postcount=161 and R (which is both can be closed from below and above, for example [0,1]) in order to conclude that you do not understand this subject.

jsfisher said:
Is it possible that 2 may not be a member of { 2 }?
You have just wrote it:

x = 2
x is_not {x}

jsfisher said:
Are 1/4 and 0.25 different numbers?
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1/4 is finite local number.

0.25 is a finite non-local number.

The organic paradigm based on non-locality\locality relation is more accurate that your context depended paradigm.

jsfisher said:
Are sets, maps, and functions all the same thing?
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1) A set is non-locality\locality relation.
2) map, function, line, logical connective, arithmetic operation, etc. are examples of the non-local aspect of non-locality\locality relation.
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PixyMisa said:
No he doesn't.


Right. He doesn't want to do mathematics. He doesn't want to learn anything. He doesn't want to be bothered with any actual effort. He just wants to be praised for his mystical insights.

He's come to the wrong forum for that.
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PixyMisa,

You simply avoid any further research of already agreed terms.

Such approach is called fanatism.
 
Little 10 Toes said:
Please explain why 1+1=2 is need of any "line\point logical connective\proposition abstraction as non-locality\locality"?
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+ is the non-local aspect of 1+1 exactly as a line is the non-local aspect of line\point or logical connective is the non-local aspect of logical connective\proposition.

Each case is nothing but some representation of non-locality\locality.
 
The Man said:
This does not help you, but only confirms the utter subjective nature of your assertions and definitions.

Why is "place" stronger then "emptiness", when it might in fact be both, a totally empty place? Just so it fits with your subjective assertions and definitions? You are the only one who seems to be avoiding the contradictions inherent in your subjective assertions and definitions.
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Emptiness or Fullness are complete (and therefore non-researchable states) and not places.

They are considered as places only in terms of non-locality\locality relation, but then the relation (which is researchable) is weaker than total fullness and stronger than total emptiness.
 
ddt said:
Fine. Then the thread really only consists of endless bickering. Time to have it off to AAH.
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This thread is not about me or about you, but about the fundamental ideas that we use in order to express ourselves.

Nobody forces you to reply here, but you.
 
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doronshadmi said:
If there is one set called D, then Cantor's proof definitely has nothing to do with X < P(X) where both of them are sets with non-finite members.
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Thank you for again demonstrating your inability to follow the simple steps in Cantor's proof.

After some initial set up to establish a meaning for a less-than relation, Cantor proceeds down a proof by contradiction approach. He assumes the existence of a set whose cardinality is not less than the cardinality of its power set. Under this assumption, he constructs a single set called D that contradicts the original assumption.

But you can't see that, can you?

You do not get that x of 0 =< x =< 1 is equivalent to D of {{}, … , {a, b, c, d, …}}, and each D (which is a member of P(X) is constructed in such a way that is not no map with any of X members.
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No, you are wrong in multiple ways. In fact, as used in Cantor's proof, the set D is proven to not exist.

But you can't see that, can you?

This result can be generalized to P(X) < P(P(X)), to P(P(X)) < P(P(P(X))) , etc. … ad infinitum.
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No. You are being sloppy with your notation for one. You are also still stuck on this comprehension failure of yours regarding the set D in Cantor's proof.

But you can't see either of that, can you?

Furthermore, P(X) members are actually constructed by this beautiful and simple

[latex]$$ D = \{ \, x \in X: x \not \in f(x) \, \} $$[/latex]​

which is not a single outcome, as you claim, but it is a general method to define P(X) out of X.
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This is another absurdity. Power set is defined well before we ever get to Cantor's proof. Nowhere in Cantor's proof is there a construction given for power set. If it were, you should be able to point directly to it, but you can't because it isn't there.

You have also completely failed to grasp what this function f(x) that appears in set D's construction has to do with anything.

Instead, you imagine things that aren't there and claim them real while completely ignoring what is there and claim them absent.


You have just wrote it:

x = 2
x is_not {x}
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I will take that to mean you are again saying 2 is not a member of {2}.

1/4 is finite local number.

0.25 is a finite non-local number.

The organic paradigm based on non-locality\locality relation is more accurate that your context depended paradigm.
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And by this, you are again saying 1/4 and 0.25 are different numbers.

1) A set is non-locality\locality relation.
2) map, function, line, logical connective, arithmetic operation, etc. are examples of the non-local aspect of non-locality\locality relation.
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Hmm, you seem to be saying the sets are different from maps and functions. So, are you retracting a previous statement of yours that sets, maps, and functions were identical, or are you merely contradicting it?


How about this one: Is a set the union of its members?
 
jsfisher said:
Thank you for again demonstrating your inability to follow the simple steps in Cantor's proof.

After some initial set up to establish a meaning for a less-than relation, Cantor proceeds down a proof by contradiction approach. He assumes the existence of a set whose cardinality is not less than the cardinality of its power set. Under this assumption, he constructs a single set called D that contradicts the original assumption.

But you can't see that, can you?
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You follow the agreed point of view of Cantor's proof, instead of to understand that D is any P(X) member that is out on the range of any X member, and as a result |P(X)| cannot be but greater than |P|.


jsfisher said:
No, you are wrong in multiple ways. In fact, as used in Cantor's proof, the set D is proven to not exist.
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Nonsense. If no D (which is any arbitrary P(X) member that is not in the range of any X member) exists, then |X| < |P(X)| can't be shown.

jsfisher said:
This is another absurdity. Power set is defined well before we ever get to Cantor's proof. Nowhere in Cantor's proof is there a construction given for power set. If it were, you should be able to point directly to it, but you can't because it isn't there.
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The result of D as an arbitrary set that is not in the range of any X member, is the result between X and another set that these arbitrary D's are its members.

One of the sets that any arbitrary D is its member is P(X) = {{}, … , {1,2,3,…}}.

You do not get that arbitrary x of 0 =< x =< 1 is equivalent to any arbitrary D of {{}, … , {a, b, c, d, …}}(which is a member of P(X)) that is constructed in such a way that it has no map with any X member.


jsfisher said:
I will take that to mean you are again saying 2 is not a member of {2}.
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2 exists both as 2 (not a member of any set) and {2} (a member of some set).


You may say: "So by your reasoning each D is not a member of any set"

My answer is: D is both not a member of any set AND also a member of some set.

One of the sets that any arbitrary D is its member is P(X).

In this case you may say: "So any member is also non-local"

My answer is: If you show that a thing is in more than a one relation, than you are talking about the non-local aspect of non-locality\locality relation.

If you show that a thing is in a one relation, than you are talking about the local aspect of non-locality\locality relation.

2 , {2} are the local aspect of 2 AND {2}.


In both cases the researchable is not less than non-locality\locality relation.


Again:

Actual Mathematics is the art of the abstraction.

Since you cannot get the notion of line\point , logical connective\proposition abstraction as non-locality\locality, you are not doin' actual mathematics.

Instead of get the general point of view of such an abstract art, you struggle here to save the context dependent dichotomist point of view of any game.

In other words, you do not understand the mathematical science as the art of the abstraction.

Actually you do your best in order to avoid the mathematical science as the art of the abstraction.

jsfisher said:
How about this one: Is a set the union of its members?
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A member is anything that inculeded in some collection where a collection is not less that non-locality\locality relation.

So a set (which is a particular form of a collection) cannot be but the union (the non-local aspect) of the members (the local aspect) of non-locality\locality relation.
 
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The Man said:
But that is just my point that polar opposite terms are just that opposites. North of the equator you are north, south of the equator you are south, as the equator is just a theoretical line so you can never be both north and south.
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Unless you are the wave aspect of the wavicle.
The Man said:
Yes this is a physical application of theoretical ascriptions, but that is exactly what doronshadmi is trying to do with his “wavicle”. If you want complimentary opposites then it is more like yin and yang or the geometric examples of infinite and finite that I gave before. They are not polar opposites but complimentary opposites that stem from each other
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Each polar is too strong or too weak as a researchable thing, only the polar's complementation is researchable.

Classical Logic and Cantor's actual infinity, is forcing the non-researchable as a researchable, which is "the mother of contradiction".
 
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nathan said:
Why is 0.25 a different number then?
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If you ignore its structure then it is also a local finite number.

Since any researchable thing is at least non-locality\locality relation, we cannot ignore the structure.

If we choose to ignore the structure, then the result is classical mathematical science, where non-locality is used as a hidden assumption and only the common local place of any result is considered.

But it is not abstraction but a trivialization of the complexity of non-locality\locality relation.

For example:

Number 1 is non-locality(a non-local atom between 0 AND 1 R members)\locality(0, 1 R members) relation.

Only locality is not researchable.

Only non-locality is not researchable.
 
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doronshadmi said:
You follow the agreed point of view of Cantor's proof, instead of to understand that D is any P(X) member that is out on the range of any X member, and as a result |P(X)| cannot be but greater than |P|.
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Yes, I understand Cantor's proof correctly as opposed to the fiction to which you adhere.

You say "D is any P(X) member that is out on the range of any X member". Members of a set do not have a range, so "range of any X member" is gibberish. Plus, you are completely ignoring all the lead up to how the set D gets defined.

The proof first assumes there is an onto mapping, f, of members of X to members of P(X). One and only one mapping is supposed, and it is not just any old arbitrary mapping. It must be onto from X to P(X).

The one and only one mapping, f, is then used to describe a set D. One set D, not many, just one. One and only one f describes one and only one D, and the construction of D is conditional on an assumption.

The set D is then used to reveal a contradiction, so the initial assumption must be rejected. Therefore, f does not exist (and the set D as well, since D relied on f).

Nonsense. If no D (which is any arbitrary P(X) member that is not in the range of any X member) exists, then |X| < |P(X)| can't be shown.
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Except, that isn't what D is. D is a single set constructed under the false assumption |X| = |P(X)|. The single set D is used to expose the assumption as false. With the assumption dismissed, D ceases to exist.

The result of D as an arbitrary set that is not in the range of any X member, is the result between X and another set that these arbitrary D's are its members.

One of the sets that any arbitrary D is its member is P(X) = {{}, … , {1,2,3,…}}.
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I see you still are a master of the gibberish.


2 exists both as 2 (not a member of any set) and {2} (a member of some set).
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Even more absurd than the original. Now you are stating four things. 2 is not a member of any set, 2 = {2}, {2} is the member of some set, and (using the transitive property for equality) 2 is the member of some set.

So, 2 = {2} and 2 is and is not the member of some set.


...you cannot get...you struggle...you do not understand...
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The comic value of this coming from someone that fails to distinguish between IF A THEN B and A ONLY-IF B propositions is priceless. No, priceless is not the right word for your words. It's similar, though. Ah, yes, I know. Worthless. Your statements are worthless, doron.


A member is anything that inculeded in some collection where a collection is not less that non-locality\locality relation.

So a set (which is a particular form of a collection) cannot be but the union (the non-local aspect) of the members (the local aspect) of non-locality\locality relation.
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Glad to see you are still clinging to this fantasy. Yeah, why let silly things like the meaning of terminology or logic interfere with a really good delusion.

Even with the following simple example that disproves your assertion, you will be unshaken in your resolve to believe in nonsense. Here goes any way:

  1. Let S be the set { {A, B}, {B, C} }
  2. The members of set S are {A, B} and {B, C}
  3. The union of the members of S is the union of {A, B} and {B, C}
  4. The union of {A, B} and {B, C} is {A, B, C}
  5. Therefore, the union of the members of S is {A, B, C}
  6. S does not equal {A, B, C}
  7. Therefore, S does not equal the union of the members of S.
See any flaws in that line of reasoning, doron?
 
I'm going to come right out and predict that he'll object to #2. In doron-speak, the members of S are {{A, B}} and {{B, C}}. Which, of course, is incorrect.
 
jsfisher said:
Yes, I understand Cantor's proof correctly as opposed to the fiction to which you adhere.

You say "D is any P(X) member that is out on the range of any X member". Members of a set do not have a range, so "range of any X member" is gibberish. Plus, you are completely ignoring all the lead up to how the set D gets defined.
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Range means that no X is mapped with any arbitrary version of D, because D is constructed not to be mapped with any X member.

The rest of your post is a direct result of your inability to get new notions and all you do is to tite yourself with your ignorence.
 
PixyMisa said:
What structure? It's a number.
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icmfig1-1.jpg

icmfig1-2.jpg

icmfig1-3.jpg

icmfig1-4.jpg

icmfig2.jpg
 
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doron, in post #1611 you write:
doronshadmi said:
Good argument Little 10 Toes.

Let us agree that only pictures or only words, are not a satisfied method for communication.
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and then post #1629 is:

doronshadmi said:
[qimg]http://www.geocities.com/complementarytheory/icmfig1-1.jpg[/qimg]
[qimg]http://www.geocities.com/complementarytheory/icmfig1-2.jpg[/qimg]
[qimg]http://www.geocities.com/complementarytheory/icmfig1-3.jpg[/qimg]
[qimg]http://www.geocities.com/complementarytheory/icmfig1-4.jpg[/qimg]
[qimg]http://www.geocities.com/complementarytheory/icmfig2.jpg[/qimg]
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only pictures! Do we see a contradiction here? Can't you even keep your thoughts straight within the timespan of 11 hours?
 
doronshadmi said:
Range means that no X is mapped with any arbitrary version of D, because D is constructed not to be mapped with any X member.
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Total gibberish - even apart from the fact that there is only one set D in Cantor's proof. Anybody with a basic understanding of (naive) set theory sees that you don't have a clue to what simple notions like range mean. I learnt those in first year high school. YMMV, but it really is no rocket science.

doronshadmi said:
The rest of your post is a direct result of your inability to get new notions and all you do is to tite yourself with your ignorence.
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This whole discussion has nothing to do with "getting new notions". It is about whether you understand actual, established mathematics. It started with my post #1583:
ddt said:
All "proofs" that Doron thus far posted here of his own volition and initiative have been found wanting to put it very mildly. You might, e.g., want to check out the thread where Doron begins with his version of a proof of Cantor's theorem that X < P(X). That "proof" sucks big time. Within the first two pages, 2 or 3 other posters - some professional mathematicians, some amateurs - take a stab at it and all give a correct rendering.
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So it's not about new notions; it's about whether you understand and can reproduce Cantor's proof. You've shown again you can't, and that you are unable to see the flaws in your own reasoning even when they're meticulously spelled out to you, in this case by jsfisher (thanks!).

Doron Shadmi, you may be a former programmer and now CAD manager at Tahal, but in mathematics you're a total crackpot. I'd say, the pinnacle of crackpottery. After posting your doronities to dozens of internet fora and basically getting the same answer, does this still not get into your head?
 
doronshadmi said:
This thread is not about me or about you, but about the fundamental ideas that we use in order to express ourselves.

Nobody forces you to reply here, but you.
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You force people to reply - to point out that your arguments are nonsense, that your mathematics is gibberish. People with less mathematical knowledge might be taken in by the first appearance that some of the things you do seem like mathematics. Hint: it's not. JREF ranks high on Google, so it's important to expose any and all cranks that come along - whether it is about homeopathy, global warming, or mathematics.

This thread is not about fundamental ideas, btw. It was about bijections, remember! And for the rest, it's about you spouting gibberish and not listening to people who know better.
 
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doron,

Since you had such trouble understanding most of my post, let's focus on just this part.

jsfisher said:
  1. Let S be the set { {A, B}, {B, C} }
  2. The members of set S are {A, B} and {B, C}
  3. The union of the members of S is the union of {A, B} and {B, C}
  4. The union of {A, B} and {B, C} is {A, B, C}
  5. Therefore, the union of the members of S is {A, B, C}
  6. S does not equal {A, B, C}
  7. Therefore, S does not equal the union of the members of S.
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See any flaws in that line of reasoning? There must be a flaw somewhere in there, otherwise a set wouldn't be a union of its members. We cannot have that, now can we?
 
There is no flaw in that reasoning.

Pedantically speaking, a set is not the union of its elements. To make the statement "a set is the union of its members" true, we must interpret this as "a set is the union of the singleton-sets that correspond to its members," i.e., X = U_{x in X} {x} = U{ {x}: x in X}. This is not the literal interpretation of that statement. Some set theories with urelements (i.e., elements that are not themselves sets) erase the distinction between an urelement and a singleton-set containing that urelement in order in certain contexts (IIRC, Quine did something like this in relation to the axiom of extensionality), but this is not usually done. There are other ways of handling urelements.

On the other hand, I can't make heads or tails of what doronshadmi is saying. There's always a bijection between any set S and itself--we can constructively build the identity map, so this is a trivial property of even ZF set theory (much less ZFC). An infinite set always has a bijection between it and a proper subset (although I think that this needs at least dependent choice if it's not taken as a definition of infinite set in the first place). What's the problem?

Perhaps it makes no sense to me because I'm coming into this thread late, but it seems to just go downhill from the get-go. Doronshadmi seems to simply not have the concept of a proof by contradiction. But this is basic logic: (p→q) ≡ (¬q→¬p).
 
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Vorpal said:
On the other hand, I can't make heads or tails of what doronshadmi is saying.

[...]

Perhaps it makes no sense to me because I'm coming into this thread late, but it seems to just go downhill from the get-go.
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Welcome to the forum, Vorpal! As to doronshadmi, you might check out his previous threads. You do risk then losing permanently 10 IQ points, though.
 
Vorpal said:
An infinite set always has a bijection between it and a proper subset (although I think that this needs at least dependent choice if it's not taken as a definition of infinite set in the first place).
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Just a minor quibble: I think you meant to qualify the proper subset has having the same cardinality as the original set.
 
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jsfisher said:
Just a minor quibble: I think you meant to say to qualify the proper subset has having the same cardinality as the original set.
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Sort of. Rather, my intended meaning is
"An infinite set always has a bijection between it and some proper subset..."
So yes, I do need to qualify that. ^_^;
[Edit] Clarification: I say "sort of" because I'd rather not say "same cardinality", as this makes the statement vacuously true. Since "same cardinality" is defined in terms of bijections, phrasing it in that was makes the statement literally mean "an infinite set always has a bijection between it and a proper subset that has a bijection to the original set."
 
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Doron, You say 1/4 and 25/100 are local but .25 is not. Can we take a simpler example. Is 1/10 local? Is .1 local? Is 1.0/10.0 local?
 
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