The power of X

By the way, the way you've defined Power(X) does not yield the same result as is outlined in the original axiom, so I don't see how any proof you base on that is valid.

Try the empty set: X = {}
Now, there are *no* mappings that will yield a set D = {} from any member of X (since there are none), thus according to that definition, Power({}) = {}

Not only does that violate what we were trying to prove, it's also wrong. Power({}) is actually {{}}
 
ArmillarySphere said:
My point was that just because any "D version (which is a Power(X) member) [...] can be mapped with any one of X
Click to expand...

Not at all. No D version (which is a subset of X and a member of Power(X)) is mapped with any one of X members, and this is exactly the reason why |Power(X)| > |X|.
 
ArmillarySphere said:
By the way, the way you've defined Power(X) does not yield the same result as is outlined in the original axiom, so I don't see how any proof you base on that is valid.

Try the empty set: X = {}
Now, there are *no* mappings that will yield a set D = {} from any member of X (since there are none), thus according to that definition, Power({}) = {}

Not only does that violate what we were trying to prove, it's also wrong. Power({}) is actually {{}}
Click to expand...

X = {} and Power(X)={{}}.

Please think general. D is not any particular result, but it is always a mamber of Power(X) that is not mapped with any member of X.

That's all. If you get it, you get what I claim.
 
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doronshadmi said:
Communication right from silence itself is the best communication, because any sound (definition) is totally clear when observed from silence itself.

No sound (definition, which is limited by nature) is a natural basis for real communication.

It can be understood only if you directly get silence itself as the simplest state of any thought.

Your struggle to find a common basis at the level of sound (definition) actually makes you tired.
Click to expand...

Are you saying Mark Twain was right?
It is better to keep your mouth shut and appear stupid than to open it and remove all doubt.

If not, what are you saying? That communication should be silent? Text-based communication is inherently silent -- unless you count the non-information-carrying noise of keyboard clatter.

Oh, and what has this to do with your contention that Cantor's proof of |X| < |Power(X)| is over-complicated?
 
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doronshadmi said:
So you did not read http://www.internationalskeptics.com/forums/showpost.php?p=3647417&postcount=70 .

If you read it, you will find that silence at its self state is the most humble state of mind, because any thought (which is limited by nature) gets its most fulfilled state if based on the simplest state of mind.
Click to expand...

Word salad.
Not even wrong.
:deadhorse

This is not mathematics.
You have been refuted on nearly EVERY SINGLE POST. Not contradicted, REFUTED. You either ignore the evidence, or simply say to re-read an earlier post. You simply (should I even use that word?) don't get it.
 
doronshadmi said:
X = {} and Power(X)={{}}.

Please think general. D is not any particular result, but it is always a mamber of Power(X) that is not mapped with any member of X.

That's all. If you get it, you get what I claim.
Click to expand...

Quite, so D is indeed *not* just any subset of X, but defined using a mapping which requires Power(X) in order to be defined at all. Which you claimed wasn't necessary before.

Still following Cantor, still using proof by contradiction, not defining Power(X) along the way as you claim. The only difference is that Cantor's version was a lot easier to follow.
 
Don At Work said:
Word salad.
Not even wrong.
:deadhorse

This is not mathematics.
You have been refuted on nearly EVERY SINGLE POST. Not contradicted, REFUTED. You either ignore the evidence, or simply say to re-read an earlier post. You simply (should I even use that word?) don't get it.
Click to expand...
Yes I was refuted by people like you that are unaware of the common basis of their thoughts, which is itself not a thought, but the simplest state of consciousness, which is the natural basis of any thought (and since a definition is some thought, it is also the natural basis of any definition).

Also a Computer stuff (or any other abstract or non-abstract mechanical method) is nothing but some agent of your consciousness.

In other words, real mathematician is first of all a person that is aware of the simplest state of consciousness as an inseparable part of his mathematical work.
 
ArmillarySphere said:
Quite, so D is indeed *not* just any subset of X, but defined using a mapping which requires Power(X) in order to be defined at all. Which you claimed wasn't necessary before.

Still following Cantor, still using proof by contradiction, not defining Power(X) along the way as you claim. The only difference is that Cantor's version was a lot easier to follow.
Click to expand...
Do you really need the axiom of the power set in order to gather infinitely many Ds (X subsets, that no one of them is mapped with any X member, because of D's definition) into a one set?


By D's definition we have D={}, D={a, b, c, d, ...} and any version between them (and no D version is mapped with any of the X members).

So, as I said, all is needed is D's definition, and we get both Power(X) and |Power(X)| > |X|, without any proof, but simply by direct construction.
 
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Firstly, assuming such sets D exist at all, which isn't clear, this would be a very roundabout way of defining the power set of X as "all possible subsets of X", i.e. the standard definition.

Secondly, how do you get to "that no one of them is mapped with any X member"? Unless you have a mapping from X to Power(X) in the first place, there's no way to get at the "remainder" set D at all.
 
ArmillarySphere said:
Firstly, assuming such sets D exist at all, which isn't clear, this would be a very roundabout way of defining the power set of X as "all possible subsets of X", i.e. the standard definition.

Secondly, how do you get to "that no one of them is mapped with any X member"? Unless you have a mapping from X to Power(X) in the first place, there's no way to get at the "remainder" set D at all.
Click to expand...

Please, please read all of http://www.internationalskeptics.com/forums/showpost.php?p=3645862&postcount=63 (including the links), it is all there.
 
doronshadmi said:
Do you really need the axiom of the power set in order to gather infinitely many Ds (X subsets, that no one of them is mapped with any X member, because of D's definition) into a one set?
Click to expand...

Finally, a genuine question.

Yes, you do. The power set axiom is independent of the other axioms of ZFC.

Here's an article discussing why in detail.


By D's definition we have D={}, D={a, b, c, d, ...} and any version between them (and no D version is mapped with any of the X members).
Click to expand...

Nope. You can't prove that without using the Power Set Axiom.
 
drkitten said:
Finally, a genuine question.

Yes, you do. The power set axiom is independent of the other axioms of ZFC.

Here's an article discussing why in detail.




Nope. You can't prove that without using the Power Set Axiom.
Click to expand...
Please provide the right link to this articale, because I do not see it.

Thank you.
 
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ArmillarySphere said:
Where does your f(x) come from?
Click to expand...


Let f be any function from X into the subsets of X according to this rule:


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


In other owrds, D's versions are exactly Power(X) and no X member is mapped with any D version.

To claim that we have no guarantee that any version of D really exists between D={} and D={a, b, c, d, …} is similar to the claim that there are R members that are greater than 0 and smaller than 1, that there is no guarantee that they are between 0 and 1.
 
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doronshadmi said:
Please provide the right link to this articale, because I do not see it.

Thank you.
Click to expand...

The link is correct; you may simply not have access to the JSTOR archives.

The paper article can be found at:

On the Independence of Set Theoretical Axioms
Alexander Abian
The American Mathematical Monthly, Vol. 76, No. 7 (Aug. - Sep., 1969), pp. 787-790

... which should be in any decent university library.

Quoting from the relevant paper (apologies for typos):

To prove that the axiom of Power-set is independent of the remaining three axioms, it is enough to consider a model whose domain consists of a single set s, where

(3) s \not\in s

i.e s is an empty set. Clearly, in this model the axiom of Extensionality is valid. However, since in this model there is no set whose element is s, the axiom of Power-set is not valid. On the other hand, since in this model Us = s, and since s is its own selection set, the axiom of sum-set as well as the axiom of Choice is [sic] valid.
Click to expand...
 
doronshadmi said:
Let f be any function from X into the subsets of X according to this rule:


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


In other owrds, D's versions are exactly Power(X) and no X member is mapped with any D version.
Click to expand...

Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

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

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?
 
jsfisher said:
How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?
Click to expand...

Oooooh, I know this one.

She's going to tell us to re-read irrelevant posts and then snidely tell us that it's an immediate consequence of some ill-defined term that we fail to understand because we are unaware of the fundamental unity of the positronium windings of the verteron coils or some such Star Trek technobabble.

And then she's going to insult people who actually read her pseudomathematics closely enough to correct some of the more gross mistakes and misstatements.
 
Doesn't the axiom merely state that "yes, every set has a power set"?

I think of the definition and axiom separately - one thing just explains what we're talking about and the other asserts some property of it. You still need the definition, however...
 
ArmillarySphere said:
Doesn't the axiom merely state that "yes, every set has a power set"?
Click to expand...

And that the power set exists and is itself a set.

See my previous posting, which cites a simple model where the "power set" does not exist (more accurately, the power set would be a proper class, not a set).
 
drkitten said:
And that the power set exists and is itself a set.

See my previous posting, which cites a simple model where the "power set" does not exist (more accurately, the power set would be a proper class, not a set).
Click to expand...

It also defines exactly what a power set is, i.e. set of all subsets of a set, doesn't it?
 
jsfisher said:
Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

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

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?
Click to expand...

Function here means mapping. So any function means any mapping from X into the subsets of X .

By using
[latex]$$ D = \{ \, x \in X: x \not \in f (x) \, \} $$[/latex]​
you constract the power set of X and also show that |Power(X)| > |X|.
 
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drkitten said:
And that the power set exists and is itself a set.

See my previous posting, which cites a simple model where the "power set" does not exist (more accurately, the power set would be a proper class, not a set).
Click to expand...
Quite, I missed that one. Fair enough - I'm not abstract enough in my thinking it seems, since I usually limit my examples to useful domains :p
 
doronshadmi said:
Function here means mapping. So any function means any mapping from X into the subsets of X .
Click to expand...
As I thought - you are using P(X), despite claiming the opposite. "The subsets of X" has no meaning unless that's what you are in fact speaking of - without the set defined as the "other end" of the mapping, there's nothing preventing all members of X to map to the same value. And as the link drkitten demonstrated, P(X) isn't even a set in the first place without the Axiom of Power Set.

Circularity, gone. What's next? Oh, right - the D

doron, you're overachieving here. It's not necessary to iterate over all possible D's here; you'd just be repeating the definition of P(X) again. All that's necessary is to show that there's a single one. But of course, that requires you to use the defined meanings of > and onto mappings... and rigorous definitions aren't really your thing, right?
 
ArmillarySphere said:
As I thought - you are using P(X), despite claiming the opposite. "The subsets of X" has no meaning unless that's what you are in fact speaking of - without the set defined as the "other end" of the mapping, there's nothing preventing all members of X to map to the same value. And as the link drkitten demonstrated, P(X) isn't even a set in the first place without the Axiom of Power Set.

Circularity, gone. What's next? Oh, right - the D

doron, you're overachieving here. It's not necessary to iterate over all possible D's here; you'd just be repeating the definition of P(X) again. All that's necessary is to show that there's a single one. But of course, that requires you to use the defined meanings of > and onto mappings... and rigorous definitions aren't really your thing, right?
Click to expand...
Since we are talking here about sets and a set is a collection of distinct members (where order is not important) then each D version is unique even if there are infinitely many mappings between X members and X subsets that their results are the same D version. In other words, we care only about the unique cases, and the collection of the unique D versions is exactly the power set of X.

There is nothing to prove here because it is a direct construction of the power set of X.

Furthermore, this construction is done in such a way that the result cannot be but
|Power(X)| > |X| and we do not have to use our brain too much in order to immediately clearly get it.

Since no set is a complete object, when compared to Total Symmetry, then the whole ugly notion of proper classes is avoided.
 
doronshadmi said:
Since we are talking here about sets and a set is a collection of distinct members (where order is not important) then each D version is unique even if there are infinitely many mappings between X members and X subsets that their results are the same D version. In other words, we care only about the unique cases, and the collection of the unique D versions is exactly the power set of X.
Click to expand...

Prove it.

There is nothing to prove here because it is a direct construction of the power set of X.
Click to expand...

No, the power set is defined by the Axiom of Power Set. You'd need to prove you've managed to enumerate the necessary ones in order to claim "the collection...is exactly the power set." by the way, you left out a very important word, union, in there.

Curiously, you had been claiming you didn't need the definition of power set; you'd define it by its construction. My, how times change.

Furthermore, this construction is done in such a way that the result cannot be but |Power(X)| > |X| and we do not have to use our brain too much in order to immediately clearly get it.
Click to expand...

Mathematical proofs clear do not depend on your use of your brain. Be that as it may, your repeated "the result cannot be but..." falls short of any sort of proof.

Since no set is a complete object, when compared to Total Symmetry, then the whole ugly notion of proper classes is avoided.
Click to expand...

Off-topic gibberish.
 
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doronshadmi said:
jsfisher said:
Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

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

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?
How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?
Click to expand...

Function here means mapping. So any function means any mapping from X into the subsets of X .

By using
[latex]$$ D = \{ \, x \in X: x \not \in f (x) \, \} $$[/latex]​
you constract the power set of X and also show that |Power(X)| > |X|.
Click to expand...


Doron, you edited my post without out any indication of what or why.

Basically, you have attributed text from a post to me, yet, that isn't my text, because you changed it. You are being very dishonest.

Why, doron? Why are you lying?
 
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What makes this great is all the whining I've ever done about graders requiring me to be very exact in my proofs and what degree of exactness I need and when can I just say, "It's trivial."

I <3 the comedy gold going on here.
 
doronshadmi said:
Since we are talking here about sets and a set is a collection of distinct members (where order is not important)
Click to expand...

Well, here's another problem. You're using naive set theory without realizing it. A set is indeed a collection of distinct members -- but not every collection of distinct members is a set. For example, the collection of all (distinct) sets is not a set, but a proper class.

You're fortunate in a sense; most people who try to apply the theorems of axiomatic set theory to informal naive concepts of sets get paradoxes. You just get gibberish.
 
jsfisher said:
It also defines exactly what a power set is, i.e. set of all subsets of a set, doesn't it?
Click to expand...

Technically, you have just offered a definition of "power set" yourself without using the power set axiom.

However, the definition by itself doesn't give you enough traction. In order to manipulate the power set of an arbitrary set X, you also need to show a few things.

First, that the power set exists.
Second, that the power set is itself a set.
Third, that the power set is unique.

The first two are given by the power set axiom. The third is given by the power set axiom plus the axiom of extensionality.

(Think of it this way : I could define Oberon as the king of the fairies, but that doesn't mean that I can do anything useful unless I can establish that fairies exist and that they have a king.)
 
drkitten said:
For example, the collection of all (distinct) sets is not a set, but a proper class.
Click to expand...
"all" means "complete" or "total".

Since a collection is not a total thing (where total is naturally undefined) "the collection of all blablabla" is nothing but a self contradiction.

Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

As long as your reasoning is closed under thoughts (where one thought holds the tail of another thought) you will not able to directly get the naturally undefined (Simplicity itself, which is the common ground of any thought, but it is not itself a thought).

The thing that is not a set, is also not a collection and it is the non-local atom which is the complete naturally undefined and unlimited common ground of any collection (finite or not, of distinct members or not).

Your circular closed under thoughts reasoning is only an illusion, and therefore it is gibberish from top to bottom.

The sad fact is the your poor students have to eat your gibberish in order to get their diploma, which is the "closed under thoughts" diploma.
 
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jsfisher said:
Doron, you edited my post without out any indication of what or why.
Click to expand...

Ho sorry, Your are right.

I edited it in order you use it as a part of my post and by mistake I also put it as a part of your quote.

Here is the right one:

jsfisher said:
Enough with your sloppy Mathematical language and hand waving. You said "let f be any function...". That means only one (although it is arbitrary as to which one). Did you really mean to speak of a family of functions, fi? That is,

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

Now, just how to construct the power set of X from the Di sets? (i.e. "D's versions")?

How are you going to show that your construction is in fact the power set of X without reference to the Axiom of Power Sets?
Click to expand...

and my answer was that "any function" means "any mapping".
 
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Here is the wiki version of the axiom of the power set ( http://en.wikipedia.org/wiki/Axiom_of_power_set ):

"Given any set A, there is a set P(A) such that, given any set B, B is a member of P(A) if and only if B is a subset of A."

drkitten said:
First, that the power set exists.
Second, that the power set is itself a set.
Third, that the power set is unique.
Click to expand...

In other words, if we have a way to construct {}, {a,b,c,d,...} and anything between them, then we get P(A), and this is exactly what Cantor did by using:
[latex]$$ D = \{ \, x \in X: x \not \in f (x) \, \} $$[/latex]​

"(Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)"

The notion of proper classes do not hold because no collection is a total (complete) thing. "The collection of all ..." DO NOT HOLD WATER, at the moment that your notion is not a "closed under thoughts" framework.
 
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doronshadmi said:
"all" means "complete" or "total".

Since a collection is not a total thing (where total is naturally undefined) "the collection of all blablabla" is nothing but a self contradiction.
Click to expand...
No, all means all. Don't introduce your vaguely defined ideolect into this.

According to your logic, the set of natural numbers does not exist, since it's the set of all natural numbers...

doronshadmi said:
Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

As long as your reasoning is closed under thoughts (where one thought holds the tail of another thought) you will not able to directly get the naturally undefined (Simplicity itself, which is the common ground of any thought, but it is not itself a thought).
Click to expand...
Incomprehensible non sequitur

doronshadmi said:
The thing that is not a set, is also not a collection and it is the non-local atom which is the complete naturally undefined and unlimited common ground of any collection (finite or not, of distinct members or not).
Click to expand...
Flatly wrong. There are a lot of proper classes - in brief, every collection that isn't a set has to be one. You yourself insisted on the collection of all distinct sets being one (which is fine, by the way - it's merely defining the domain).

doronshadmi said:
Your circular closed under thoughts reasoning is only an illusion, and therefore it is gibberish from top to bottom.

The sad fact is the your poor students have to eat your gibberish in order to get their diploma, which is the "closed under thoughts" diploma.
Click to expand...
Yes, how horrid that students have to learn to explain their reasoning...
 
ArmillarySphere said:
No, all means all. Don't introduce your vaguely defined ideolect into this.

According to your logic, the set of natural numbers does not exist, since it's the set of all natural numbers......
Click to expand...

Don't use your "closed under thoughts" circular reasoning, in order to understand my framework.

According to my Logic there exists the set of Natural numbers, but like any collection, it is incomplete (something that cannot be understood by a "closed under thoughts" circular reasoning).
ArmillarySphere said:
Incomprehensible non sequitur
Click to expand...
You are right, because your abstract ability is "closed under thoughts" .
ArmillarySphere said:
Yes, how horrid that students have to learn to explain their reasoning...
Click to expand...
Very simple. Any student is first of all directly aware of the simplest state of consciousness, which is itself not a thought but the unbounded atom naturally undefined common ground of any thought (and again, any definition is first of all a thought).

By using the non-researchable as the common ground of the researchable we get a circular-free framework.

This simple beauty cannot be understood by any person that uses a "closed under thoughts" circular reasoning.

By using a circular-free framework, every student will be able the share his unique point of view in such a way that will be understood by the rest of the students, because they share the same common ground, which is the natural basis of any thought (any definition).

I know that you do not understand a single word of what I say because you try to understand it by a "closed under thoughts" method.
 
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doronshadmi said:
This time please read http://www.internationalskeptics.com/forums/showpost.php?p=3648068&postcount=97 until the end of it.

Thank you.
Click to expand...

Oooh, can I call it or what?

I wrote:

She's going to tell us to re-read irrelevant posts
Click to expand...

Which she did:

This time please read http://www.internationalskeptics.com/forums/showpost.php...8&postcount=97 until the end of it.
Click to expand...


I then wrote that :
and then snidely tell us that it's an immediate consequence of some ill-defined term
Click to expand...


and she then wrote:
"all" means "complete" or "total".

Since a collection is not a total thing (where total is naturally undefined) "the collection of all blablabla" is nothing but a self contradiction.
Click to expand...

I then predicted that she would say ...
that we fail to understand because we are unaware of the fundamental unity of the positronium windings of the verteron coils or some such Star Trek technobabble.
Click to expand...

just as she did here:

By using the non-researchable as the common ground of the researchable we get a circular-free framework.

This simple beauty cannot be understood by any person that uses a "closed under thoughts" circular reasoning.

By using a circular-free framework, every student will be able the share his unique point of view in such a way that will be understood by the rest of the students, because they share the same common ground, which is the natural basis of any thought (any definition).

I know that you do not understand a single word of what I say because you try to understand it by a "closed under thoughts" method.
Click to expand...

and finally, I predicted that
And then she's going to insult people who actually read her pseudomathematics closely enough to correct some of the more gross mistakes and misstatements.
Click to expand...

Which she fulfilled by writing

Your limited notion is based on limited notions. It is circular because you use thoughts in order to define thoughts (any definition is first of all a thought).

As long as your reasoning is closed under thoughts (where one thought holds the tail of another thought) you will not able to directly get the naturally undefined (Simplicity itself, which is the common ground of any thought, but it is not itself a thought).

[snip]

Your circular closed under thoughts reasoning is only an illusion, and therefore it is gibberish from top to bottom.

The sad fact is the your poor students have to eat your gibberish in order to get their diploma, which is the "closed under thoughts" diploma.
Click to expand...

Can I call it, or what? I'd like my million now, Mr. Randi....
 
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