Cont: Deeper than primes - Continuation 2

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doronshadmi said:
Dear jsfisher,

This is my last post to you on this fine subject.
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Thank you. Too bad you could not define your terms, though.

Fortunately Mathematics is not restricted only to your arbitrary |N| observation.
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I have no such arbitrary observation, nor would Mathematics be restricted by it. Neither your re-definitions nor your straw change the value of 0.999....
 
http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110 is aimed to those who are interested in the discussed fine subject, which is rigorously defined by the abstract mathematical fact of the strict difference among carnality |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...

Using ∞ (as currently done among mathematicians) in order to deduce conclusions in terms of infinity is not accurate enough, simply because it does not use the well established mathematical proof of |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... different levels of infinity, where each one of them is defined in no more than one step (no process of more than one step is involved) .
 
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doronshadmi said:
http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110 is aimed to those who are interested in the discussed fine subject, which is rigorously defined by the abstract mathematical fact of the strict difference among carnality |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...
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You keep using those words, but do not understand what they mean.

Using ∞ (as currently done among mathematicians) in order to deduce conclusions in terms of infinity is not accurate enough...
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No, it is you who do not understand how infinity is used in something like 0.999.... (In fact, it isn't used at all, but that nuance escapes you, doesn't it.) As a result, you make things up in support of your own confusion, and you erect strawmen in attempts to discredit the actual meaning of things.

The things you make up, you cannot ever define, either. That indicates that you, yourself, have no idea what you are saying.

...simply because it does not use the well established mathematical proof of |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... different levels of infinity, where each one of them is defined in no more than one step (no process of more than one step is involved) .
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There's another phrase you misuse and don't understand.
 
What? All the other posters already left Doron? I can understand that; all the talk in the world of being peaceful does not balance the acts of strife and opposition that Doron really commits.

Doron, again. If you are in a discussion, you *MUST* listen to the other side and you *MUST* concede when you are *WRONG*.

If you are unable to do so, then on *any* board on the Internet, people will simply ignore you.

If you look at the preceding weeks, you will find that, with the exception of Apathia, each and everyone, in just about any tone of voice possible, from friendly to arrogant, has asked you the same same thing:

- Define your concepts. Not *show* your concepts, which you keep doing, but *defining* them. Put borders around them, explain why these borders are valid etc.

Whatever you think, whatever you say, if you can not participate in a discussion, then in a few years, when you are gone from the planet, everything you have ever said is just stored and ignored. It won't have mattered that you existed.

To change that, participate, not just direct, and concede when you are wrong.
 
Dear Dessi,

Let's simplify http://www.internationalskeptics.com/forums/showpost.php?p=10334974&postcount=158 as follows:

The serial solution: |1| worker puts |N| stones along an infinite road by speed |N|.

The parallel solution: |N| workers put |N| stones along an infinite road by speed |1|.

So, in both cases the mission is accomplished by one step (|1| worker with speed |N| = |N| workers with speed |1|).

By using one step for each cardinal number of the forms |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... it is clear that no mission of lower cardinality is accomplished if observed from higher cardinality, or in other words, the power of lower cardinality is insignificant in order to reach the power of higher cardinality.
 
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doronshadmi said:
Dear Dessi,

Let's simplify http://www.internationalskeptics.com/forums/showpost.php?p=10334974&postcount=158 as follows:

The serial solution: |1| worker puts |N| stones along an infinite road by speed |N|.

The parallel solution: |N| workers put |N| stones along an infinite road by speed |1|.
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Do you know what these two processes have in common with the evaluation of 0.999...? Absolutely nothing. Why do you keep bringing it up?

Do you know what's curious about your so-called solutions (aside from their irrelevance to the infinite series topic)? You so desperately try to make aleph-null behave like an integer, which it isn't, so they are mathematically meaningless.

All that aside, though, why haven't you found a place for aleph-one in any of this? Considering how much mathematics has been ignored to this point, forcing in aleph-one and some of its higher-numbered friends should be easy for you.
 
jsfisher said:
You so desperately try to make aleph-null behave like an integer
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Not even close.

You are simply missing the one step notion, no matter what cardinality>0 (finite or infinite) is used.

As a result
doronshadmi said:
By using one step for each cardinal number of the forms |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... it is clear that no mission of lower cardinality is accomplished if observed from higher cardinality, or in other words, the power of lower cardinality is insignificant in order to reach the power of higher cardinality.
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is not in your scope.

Here is a concrete example (your rhetoric question) that supports my argument about you:
doronshadmi said:
Do you know what these two processes have in common with the evaluation of 0.999...? Absolutely nothing.
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It is clearly seen that you are totally missing the fact that there is one step in both parallel and serial solutions, no matter what cardinality>0 (finite or infinite) is used for a given solution.

Moreover, you are totally missing the fact that a given solution in a given cardinality>0 is not satisfied if observed from higher cardinality (as written in my first quote in this post).

Generally your mathematical universe is the result of no more than one level for each observation (you systematically avoiding observations of lower levels from higher levels, so there is no wonder that my theorem is not in your scope).
 
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doronshadmi said:
Not even close.
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Perhaps you don't understand your own posts. Your wrote, among other things, "|N| workers". You are, in fact, using aleph-null as if it were an integer. I am even more than close; I am dead on.
 
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doronshadmi said:
Please change "|N| workers" to "|N| integers", and walla, you have no argument.
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No, the same argument remains: Whether workers or integers, you are still trying to use |N| as if it were an integer.

Be that as it may, it still has no relevance to the valuation of 0.999....

(Walla?? Half a city in Washington State?)
 
doronshadmi said:
Wrong, there is a collection of |N| integers, where |N| is not one of them.
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There are infinitely many integers and the cardinality of the set of integers is aleph-null. However, it is incorrect to say the number of integers is aleph-null or anything semantically equivalent, including "|N| integers". As a colloquial convenience, the phrase and its semantic equivalents, "the number of integers is infinite", may be used, but never as a prelude to using 'inifinity' as an ordinary number.

Be that as it may, it is still irrelevant to the valuation of 0.999....
 
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doronshadmi said:
Your inabilities are clearly demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=10349404&postcount=208.
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Ah, another Doron-classic massively re-edited post!

One step or a billion (or, dare I say it, infinitely many steps), it is still a process you are chasing. Moreover, whether the steps are performed sequentially or all in parallel, or any combination of the two, it is still the same number of steps. But what of that, either way, you remain fixated on process.

And it still is irrelevant to the valuation of 0.999.... No process, no algorithm, just a simple limit. So simple that only rational numbers are needed. The rest of the reals (and most* of the rationals for that matter) are never needed to establish 0.999... as identical to 1.

The only way for you demonstrate that 0.999... and 1 are not identical is to show that 1 does not satisfy the N/epsilon requirement for the limit of the partial summations corresponding to 0.999....


[SIZE=-1]*"Most" takes a figurative meaning in this parenthetical remark.
[/SIZE]
 
Ah, I see we are back to Doron being snide and insulting again...

I wonder what has happened to the Doron that all of his friends and family said to be a peace-loving person.

Doron, how is communicating with others working out for you? What? Everyone left already?
 
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doronshadmi said:
This is another example that supports my argument about your one-level mathematical universe, as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=10349404&postcount=208.
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Doron, you do not get to redefine the meanings of things to conform to your misunderstandings. The valuation of 0.999... stands at 1. Your attempts to distract with irrelevant references to power sets doesn't change that.


(Curious, though. You used to reject Cantor's Theorem. It was just wrong; Cantor was just wrong. Now, you embrace it, even though it lacks utility for your purpose. How flexible of you.)
 
jsfisher said:
The valuation of 0.999... stands at 1.
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If observed only from |N|.

http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110 is simply not in your |N|_only scope.

jsfisher, you simply have no argument anymore, so you are digging in the past and desperately try to convince yourself that real mathematics fits to your |N|_only scope.

jsfisher said:
Doron, you do not get to redefine the meanings
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I do not redefine anything. My argument is true if |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... is true.

Simple as that, as clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=10343813&postcount=202.

Your ∞ hands waving does not hold water.
 
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jsfisher said:
Doesn't matter how it is "observed." Doron, you don't get to redefine mathematics to accommodate your personal confusion.
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Dear jsfisher.

This is the beautiful thing here, I do not redefine anything.
 
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doronshadmi said:
Dear jsfisher.

This is the beautiful thing here, I do not redefine anything.
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Sure you did. The decimal notation 0.999... has a well-defined meaning, that of a series. The series has a value determined by the limit of the partial summation sequence corresponding to the series. The determination of limits is well-defined.

The consequence of all that is that 0.999... is identical in value to 1.

Your attempt to "observe" it differently is a redefinition. I can observe 4 differently as 5; that doesn't make it 5. Moreover, your appeal to cardinal numbers is unwarranted since they have no function in establishing the value of 0.999....
 
jsfisher said:
Sure you did.
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Sure I did not.

jsfisher said:
Your attempt to "observe" it differently is a redefinition.
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Not at all, it is based on the well defined mathematical fact that |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...

jsfisher said:
I can observe 4 differently as 5; that doesn't make it 5.
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Even in this simple case you fail. Observing 4 from 5 simply enables one to know that 5 > 4. There is nothing in this observation that can be interpreted as if 4 is 5.

jsfisher said:
Moreover, your appeal to cardinal numbers is unwarranted since they have no function in establishing the value of 0.999....
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They have a function in establishing the value of 0.999..., but it clearly not in the scope of one that interpretes 4 as 5.
 
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doronshadmi said:
Not at all, it is based on the well defined mathematical fact that |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...
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You keep repeating that, but you never actually show how it (whatever "it" actually is) is based on those inequalities. How do you connect Cantor's Theorem to the definition for limits in such a way that 0.999... and 1 are not identical?
 
jsfisher said:
You keep repeating that, but you never actually show how it (whatever "it" actually is) is based on those inequalities.
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All you have to do is to deduce a given mathematical framework in terms of cardinality, in order to provide a given solution.

For example:

Finite or infinite cardinality:

A given solution that is satisfied by at least some cardinality > |1|, is not satisfied by some cardinality ≤ |1|.

Infinite cardinality:

A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.

A given solution that is satisfied by at least some cardinality > |P(N)|, is not satisfied by some cardinality ≤ |P(N)|.

A given solution that is satisfied by at least some cardinality > |P(P(N))|, is not satisfied by some cardinality ≤ |P(P(N))|.

A given solution that is satisfied by at least some cardinality > |P(P(P(N)))|, is not satisfied by some cardinality ≤ |P(P(P(N)))|.

etc. ... at infinitum.

jsfisher said:
How do you connect Cantor's Theorem to the definition for limits in such a way that 0.999... and 1 are not identical?
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A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.
 
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doronshadmi said:
jsfisher said:
You keep repeating that, but you never actually show how it (whatever "it" actually is) is based on those inequalities.
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All you have to do is to deduce a given mathematical framework in terms of cardinality, in order to provide a given solution....
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So you are unable to make clear whatever "it" actually is.

Telling us all what it's based on and could be deduced from is just hand-waving without ever defining "it".

jsfisher said:
How do you connect Cantor's Theorem to the definition for limits in such a way that 0.999... and 1 are not identical?
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A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.
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And this is relevant to the definition for limits just how?
 
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doronshadmi said:
A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.
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I am not entirely sure why the at least is emphasized by an underscore...

The greater than symbol already signifies this.

And as for the logic... this is the same as stating: If it is not black, then it is a different color...
 
realpaladin said:
I am not entirely sure why the at least is emphasized by an underscore...

The greater than symbol already signifies this.

And as for the logic... this is the same as stating: If it is not black, then it is a different color...
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And you aren't even going to ask what it means to satisfy a solution, now are you? Nor how a solution might be satisfied by a cardinal number, either, right?

;)
 
jsfisher said:
So you are unable to make clear whatever "it" actually is.

Telling us all what it's based on and could be deduced from is just hand-waving without ever defining "it".
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This "it" is simply cardinality, and it measures the minimal needed values that satisfy a given value, for example:

0.9+0.09+0.009 that has |3| values can't satisfy value 1, and in this case at least 0.9+0.09+0.009+... that has |N|+1 = |N| values, satisfies value 1.

In case that the minimal needed values that satisfy 1 is at least |P(N)|+1 = |P(N)|, 0.9+0.09+0.009+... that has |N|+1 = |N| values, can't satisfy value 1.

In case that the minimal needed values that satisfy 1 is at least |P(P(N))|+1 = |P(P(N))|, no |P(N)|+1 = |P(N)| of such values satisfy value 1.

Etc. ... ad infinitum.

jsfisher said:
And this is relevant to the definition for limits just how?
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It is about the ability to satisfy a given value, as explained above.
 
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doronshadmi said:
This "it" is simply cardinality, and it measures the minimal needed values that satisfy a given value
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Just how do values satisfy a given value? What values would satisfy 42, just as an example?

for example:

0.9+0.09+0.009 that has |3| values
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No, that is an expression of three values (and, no, I don't need to extract the absolute value of 3). The expression has a value, and that is 0.999.

...can't satisfy value 1
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"Satisfy value 1"? Perhaps you meant "equal" in place of "satisfy value"?

...and in this case at least 0.9+0.09+0.009+... that has |N|+1 = |N| values, satisfies value 1.
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0.9 + 0.09 + 0.009 + ... is an infinite series. The only thing important in determining the value of the series is the limit of the related sequence of partial summations. No infinity is needed to establish that the limit is 1 and thus 0.999... is identical to 1.

(By the way, your assertion of |N| + 1 terms are involved is just bizarre. So is your usage of the phrase, "at least".)

In case that the minimal needed values that satisfy 1 is at least |P(N)|+1 = |P(N)|, 0.9+0.09+0.009+... that has |N|+1 = |N| values, can't satisfy value 1.
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And what case would that be? Again, series, limits, value without every visiting any flavor of infinity.

...
It is about the ability to satisfy a given value, as explained above.
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Your use of that highlighted word is novel.
 
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jsfisher said:
(By the way, your assertion of |N| + 1 terms are involved is just bizarre.
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The bizarre thing here is actually your mathematical framework that excludes the different values of cardinality as an essential factor of a given solution.

As a result http://www.internationalskeptics.com/forums/showpost.php?p=10318337&postcount=7, http://www.internationalskeptics.com/forums/showpost.php?p=10318659&postcount=9, http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110 and http://www.internationalskeptics.com/forums/showpost.php?p=10353611&postcount=233 are not in the scope of your mathematical framework.

Moreover, the treatment of your framework about infinity (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=10344228&postcount=203) simply excludes the different values of transfinite cardinality as an essential factor of a given solution.

Since your framework excludes the different values of cardinality as an essential factor of a given solution, there can't be any communication between us about this fine subject.

You can enjoy your ∞ hand-waving as much as you like, which uses each cardinal number separately from the other cardinal numbers, but you will not find my framework under your hand-waving.

Here is a concrete example of your hand-waving, by using your own words
jsfisher said:
0.9 + 0.09 + 0.009 + ... is an infinite series. The only thing important in determining the value of the series is the limit of the related sequence of partial summations. No infinity is needed to establish that the limit is 1 and thus 0.999... is identical to 1.
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that simply excludes the different values of cardinality (and in this case, transfinite cardinality) as an essential factor of a given solution.
 
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http://www.internationalskeptics.com/forums/showpost.php?p=10316119&postcount=4298 is defined by using different values of cardinality as an essential factor of a given solution, and it is definitely not in the scope of a framework that uses only each cardinal number separately from the other cardinal numbers, in order to provide a given solution.

jsfisher said:
0.9 + 0.09 + 0.009 + ... is an infinite series.
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More accurately, 0.9 + 0.09 + 0.009 + ... is an infinite series of countable |N| values and it is < 1 if it used among an infinite series of uncountable |P(N)| values.

In order to understand it all you have to do is to realize, for example, how 0.9 + 0.09 + 0.009 finite series of countable |3| values is < 1 if it used among an infinite series of countable |N| values like 0.9 + 0.09 + 0.009 + ...

The same principle holds for both cases.
 
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jsfisher said:
0.999... stands at 1.
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Just to be technically accurate, the limit of 0.999... stands at 1. No matter how many 9's you add to the string, the sum will never add up to 1.

0.999... is a convergent countably infinite series. Trying to make more of it than that like doronshadmi is doing or making an unprovable assertion that an infinite string of 9's would still add up to < 1 is to just generate meaningless confusing jargon.
 
doronshadmi said:
You can enjoy your ∞ hand-waving as much as you like, which uses each cardinal number separately from the other cardinal numbers, but you will not find my framework under your hand-waving.
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Some true words by Doron... I thoroughly enjoy the rigorously proven thing that he denotes by 'hand-waving', and I sure have not found even the cornerstone of a framework by Doron in any of the 10k+ posts in over 7 years.

Doron, again, it is not the fault of the students (there have been so many, with so many different views; from Apathia who wishes you the best but does not understand your words, via Dessi who is your superior in mathematics but still can't follow you, to JsFisher and myself who are just tenacious in trying you to commit to something that is defined), but the fault of the teacher if the knowledge is never passed on.

Because even *if* we all were to agree you are right, you just wait until there is something to kibitz about; you think that discussion and kibitzing makes you look smart... it does not, it makes you look quarrelsome and angry.
 
doronshadmi said:
My terms are rigorously defined by the fact that |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...
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As your diagram in http://www.internationalskeptics.com/forums/showpost.php?p=10334974&postcount=158 showed, this is true only as long as N is some finite subset of the set of natural numbers. You haven't demonstrated this to be true for a countably infinite set of natural numbers.

I'd go as far as to say that if N is the set of all natural numbers, then all of the supersets that you generate are also countably infinite.
 
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