Deeper than primes - Continuation

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jsfisher said:
The parenthetical comments in italics are your fantasy, not anything in ZFC. You simply are making stuff up, as you so frequently do.
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jsfisher, fantasy is only at the subjective level of existence, which is dependent on the objective level of existence (which is independent on any given identity, or fantasy).

By using again and again "making stuff up", "fantasy" etc., I think that this is your way to describe the non-useful aspect of the subjective (b.t.w identity is a very useful aspect of the subjective, as seen, for example, in ZFC).

Your reasoning simply can't deal with the objectivity of the discovered, which is independent on the subjectivity of the invented.

jsfisher said:
You post also relates in no way to the post of mine you quoted. So there is no evidence you understood my post at all. This lends further support to the conclusion your reading comprehension issues are at the root of why you fabricate instead of understand.
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There is a clear evidence that can't grasp the objective level of set's existence, which is independent on any given identity exactly because identities are subjective inventions.

jsfisher said:
Also, I note your continued parroting of "S(y)" in your posts as if you believed it has meaning stripped from its original context. Even more evidence of failures in understanding.
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Nothing was stripped from its original context. On the contrary, I address the dependency of the invented (the subjective level of members like "S(y)") on the discovered (the objective level of set's existence ("there exists set X")).

In ZFC there is an hierarchy of dependency (which means that both X existence and X identity are used), where X identity depends on X existence, but X existence does not depend on X identity.

For example, nothing was stripped or "slightly hacked" from ZFC by post http://www.internationalskeptics.com/forums/showpost.php?p=10017698&postcount=3731, only in your one level imagination.

jsfisher said:
ETA:
By the way, in what way is "the empty set {}" different from "the empty set" or from "{}"? Or do you just feel compelled to restate things redundantly by repeating things?
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My mistake (I used what is written in http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory without first checking it) the correct one is "the empty set ({})". Thank you for your remark.
 
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doronshadmi said:
...<non sequitur sequence snipped>...

Nothing was stripped from its original context. On the contrary, I address the dependency of the invented (the subjective level of members like "S(y)") on the discovered (the objective level of set's existence ("there exists set X")).
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It surprises me not at all that you cannot see that your continual re-presentation of "S(y)" is without its context.

In ZFC there is an hierarchy of dependency (which means that both X existence and X identity are used), where X identity depends on X existence, but X existence does not depend on X identity.
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And, yet, you cannot point to that anywhere in the ZFC axioms.

...
My mistake (I used what is written in http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory without first checking it) the correct one is "the empty set ({})". Thank you for your remark.
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You must get a different version of Wikipedia in Isreal. What does it show for Axiom of infinityWP?
 
Doron,

just waiting for a long time and then suddenly starting all over does not work.

You still fail to show how any of this connects with the questions stated before you rebooted this thread...

And really, if I would accept your 'reboot' statement, can you show how this helps in your roadmap, or in the two islands thought experiment?
 
realpaladin said:
@JSFisher: what is the angle this time?
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One of his favorites: There really is no infinity, except there sort of is, but you never get there, because everything is a process, and you can always take one more step, and by God! those braces have feelings, too!

And anyone who thinks otherwise doesn't get it.
 
jsfisher said:
It surprises me not at all that you cannot see that your continual re-presentation of "S(y)" is without its context.
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It surprises me not at all that you cannot see that "S(y)" is an invention that is dependent on the discovered, but not vice versa.

jsfisher said:
And, yet, you cannot point to that anywhere in the ZFC axioms.
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And, your one level reasoning is not useful in order to point to that anywhere in the ZFC axioms, so?

jsfisher said:
You must get a different version of Wikipedia in Isreal. What does it show for Axiom of infinityWP?
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http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory is the English version of Wikipedia all over the internet.

Moreover, in http://www.internationalskeptics.com/forums/showpost.php?p=10011797&postcount=3709 you use an expression that appears in http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory, so for better communication with you I used what comes before this expression, which is:

"...there exists a set X such that the empty set Ø is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X."

So, the mistake is in the English version of Wikipedia all over the internet, but your reply about this case clearly expose how you actually ignore the details my replies, and in this case my reply was
doronshadmi said:
My mistake (I used what is written in http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory without first checking it) the correct one is "the empty set ({})". Thank you for your remark.
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that simply ignored by you.

No wonder that by systematically using such an attitude, you reinforce your misunderstanding of my replies.
 
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jsfisher said:
One of his favorites: There really is no infinity, except there sort of is, but you never get there, because everything is a process, ...
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This is another demonstration of how you systematically ignore what I write (http://www.internationalskeptics.com/forums/showpost.php?p=10023528&postcount=3756):
doronshami said:
It has to be stressed that level of members is an invention that is defined in some moment in time, but it does not mean that, for example, "whenever a set y is a member of X, then S(y) is also a member of X" is a process in time, it simply demonstrates the inaccessibility of the invented (subjective) level of members of a given set, to the discovered (objective) level of set.
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No wonder that you have no clew of what I am talking about, and you are not in any position to explain what I write (except in your own fantasies).
 
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doronshadmi said:
...<non sequitur sequence snipped>...

http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory is the English version of Wikipedia all over the internet.
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Yes, I know. My comment was steeped in sarcasm. If you look carefully at what is written, not simply what you imagine you'd like to be there, the character sequence "{}" does not appear where you claimed.

More lack of attention to detail on your part, Doron. It only undermines your position.
 
"There exists ..." is the discovered (objective) level of existence, which is independent on the invented (subjective) level of existence.

In other words, identification (which is done at the invented (subjective) level of existence) needs the discovered (objective) level of existence, in order to exits, but the discovered (objective) level of existence does not need identification in order to exists.

This notion of shown in the following:

Now, by following these notions as the basis of cardinal numbers (which are based on ZFC set-theory level, but they are not at ZFC set-theory level), the right way to express cardinal numbers is shown in the following example:

{||}=0 , {|{}|}=1 , {|{},{{}}|}=2 etc. , or {|{}, {{}}, {{},{{}}}, ... |} < aleph0 (where aleph0 is at the objective level (notated here by the outer "{" and "}"), where no number of members (where members are at the subjective level) is accessible to it).

As can be seen, the subjective (invented) level of members is inaccessible to the objective (discovered) level of a given set (where this objectivity is expressed by the outer "{" and "}", whether a given set is empty or non-empty).

In other words, given a set with infinitely many members, they are inherently under construction (where "inherently under construction" is not a processes) (for example: "whenever a set y is a member of X, then S(y) is also a member of X") since the subjective level of members (which provide the identity of a given set) is inaccessible to the objective level of a given set (where this objectivity is expressed here by the outer "{" and "}", whether a given set is empty or non-empty).

More generally, Mathematics (abstract or non-abstract) is useful exactly because it is discovered and invented, where the invented depends on the discovered, but not vice versa.
 
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So, Doron, at which point do we get somewhere in the roadmap from this to the world-peace?

Again, it appears that all Organic Mathematics has to offer is strife and misunderstanding.

But I probably misunderstand all of that ;)
 
doronshadmi said:
This is another demonstration of how you systematically ignore what I write (http://www.internationalskeptics.com/forums/showpost.php?p=10023528&postcount=3756):

No wonder that you have no clew of what I am talking about, and you are not in any position to explain what I write (except in your own fantasies).
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Doron, first: "attack the argument, not the arguer" as per contract with this forum.

Second, if JSFisher is in no position to explain what you write, then please be so civil to refrain from taking a position that explains what he writes, whether it be his fantasies or not.

Finally, where in ZFC, I use the Wolfram pages; they are better suited for mathematics than Wikipedia, can you find any support for your claims? Here is the link: http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
 
"There exists AND does not exist set X" is a contradiction at the objective (platonic) level of discovery (or in other words, the objective (platonic) level is not discovered, which has no impact on the objective (platonic) existence of set X).

"There exists set X" such that "it is AND it is not its own member" is a contradiction at the subjective (non-platonic) level of invention, which has an impact on X's subjective invented identity, but it does not have any impact on X objective level of discovery (X exists as a platonic object even if its non-platonic subjective level is not well-defined).

By understanding this hierarchy of dependency, any Mathematics that is based on sets, is useful exactly because it is discovered (this is its objective platonic level) AND invented (this is its subjective non-platonic level), such that the invented depends on the discovered, but not vice versa.

The non-platonic subjective level of set is inaccessible to the platonic objective level of set, where this inaccessibly provides the room for unlimited subjective non-platonic interpretations (inventions) that are based on the platonic objective level of set, but not vice versa.

More details are given, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=10025797&postcount=3773.
 
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doronshadmi said:
"There exists AND does not exist set X" is a contradiction at the objective (platonic) level of discovery (or in other words, the objective (platonic) level is not discovered, which has no impact on the objective (platonic) existence of set X).
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"There exists a set X" is not a complete statement. Neither is its complement; neither is the conjunction of the two. You cannot correctly say the conjunction is a contradiction simply because you never got far enough constructing the statement.

On the other hand, the statement, "for some X, X is a set" is a complete statement (a true one, in fact), and the statement, "for some X, X is a set and X is not a set" is a false (complete) statement. There are no levels to it being false, though.

"There exists set X" such that "it is AND it is not its own member" is a contradiction at the subjective (non-platonic) level of invention, which has an impact on X's subjective invented identity, but it does not have any impact on X objective level of discovery (X exists as a platonic object even if its non-platonic subjective level is not well-defined).
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Curious use of quotations marks, Doron. Do you not understand the meaning they convey? At any rate, the statement, "there exists set X such that X is a member of X and X is not a member of X", is a false statement. There are no levels to it being false.

If there were levels, then you would be able to identify two such levels wherein the truth value for the statement differ.
 
jsfisher said:
"There exists a set X" is not a complete statement.
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"There exists a set X" is a complete objective platonic level of set X, which is discovered.

"There exists AND does not exist a set X" is contradiction that prevents the discovery of the complete objective platonic level of set X.

Yet, platonic objects exist independently of their discovery.

"there exists set X (the objective platonic level of discovery) such that (the non-platonic level of invention) X is a member of X and X is not a member of X", is a false statement at the subjective level of non-platonic invention.

jsfisher said:
If there were levels, then you would be able to identify two such levels wherein the truth value for the statement differ.
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The truth value at the objective platonic level of set X (which is discovered) is only tautology, and it is indeed different from the truth value at subjective non-platonic level members (which is invented), which is not only a tautology.

The discovered is independent on the invented, but the invented depends on the discovered.

Your one level reasoning is not useful in order to deduce the difference between the discovered and the invented.

As a result you can't be aware of the inaccessibility of the invented to the discovered, as deduced by the following example:

{||}=0 , {|{}|}=1 , {|{},{{}}|}=2 etc. , or {|{}, {{}}, {{},{{}}}, ... |} < aleph0 (where aleph0 is at the objective level (notated here by the outer "{" and "}"), where no number of members (where members are at the subjective level) is accessible to it).

The outer "{" and "}" notates the notion of actual infinity, which is at the complete objective platonic level of set X, whether X has or does not have members, and if X is non-empty, then no number of its members (which are at the subjective non-platonic level of invention) is accessible to the level of actual infinity.

The
 
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jsfisher said:
No, according to ZFC.
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jsfisher, this is no more than your interpretation of ZFC, which unable to deduce the difference between the platonic and the non-platonic, and how they are usefully used by ZFC.
 
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doronshadmi said:
...platonic objects...
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So, despite all the pretense of discussing set theory and other actual mathematics, you really are just wallowing in Shadmized philosophy.

I will sleep better tonight now knowing all the defects in Mathematics you have alleged are actually just your philosophic musing. Mathematics remains unblemished by your assault.
 
doronshadmi said:
According to his one level reasoning, so?
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No, according to the strict and defined definition of ZFC.

It is time to fire up a new 'Doron Shadmi's errors' resumé I guess.

Doron, if you augment your reasoning statements in a way that makes it clear that *you* need to modify them so *your* theory works, then we all *have* to wait and see where you take us.

But as long as you try to make definitions of others seem 'broken', we can do naught but correct you...

And all of this is just rehashing the rehash that was rehashed 7 years ago.

We, the few that stick to this thread, have seen just about every argument you have made and we are eager to see where it will take us.
 
jsfisher said:
Mathematics remains unblemished by your assault.
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Wrong, Mathematics is based on Philosophy, whether you ignore it or not.

"There exists set X" is only a tautology, where what comes after "such that" is not only a tautology.
 
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doronshadmi said:
jsfisher said:
So, despite all the pretense of discussing set theory and other actual mathematics, you really are just wallowing in Shadmized philosophy.

I will sleep better tonight now knowing all the defects in Mathematics you have alleged are actually just your philosophic musing. Mathematics remains unblemished by your assault.
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Wrong, Mathematics is based on Philosophy, whether you ignore it or not.

"There exists SET X" is only a tautology, where what comes after "such that" is not only a tautology.
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Amazing... Doron manages to write a response that almost, but not entirely, completely shoots past the target...

Doron, JSFisher never said what your response alludes to.

I for one would welcome some rigour in your replies.
 
jsfisher said:
Curious use of quotations marks, Doron. Do you not understand the meaning they convey? At any rate, the statement, "there exists set X such that X is a member of X and X is not a member of X", is a false statement. There are no levels to it being false.
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Let's use Cantor's Theorem in order to explicitly demonstrated the difference between the platonic level of existence and the non-platonic level of existence.

A = {a, b, c, ...}

P(A) = {{}, {a, b, c, ...}, ...}

By Cantor's theorems its is proved that |A|<|P(A)|, by using two steps:

1. |P(A)| can't be < |A| (for example, there are at least such pairs: (a,{a}), (b,{b}), (c,{c}), ...).

2. There is at least one member of P(A) (let's call it S), which is defined as "the set of all A members, which are not members of the P(A) members that are paired with them", so given some A member (let's call it J) to be paired with S, if J is not a member of S, then according to S definition, J is a member of S, but then according to S definition, J is not a member of S.

Since S is well-defined and since J membership with S leads to contradiction (J is a member AND not a member of S), it is concluded that S is not paired with any A member, which enables to prove that |P(A)|>|A|.

In other words, J can't be defined as some A member that is paired with S (which is a well-defined P(A) member).

Yet, the platonic objective level of existence (notated by the outer "{" and "}" in both sets) is independent on the non-platonic level of existence of the members these sets, and this notion is expressed in terms of cardinality as follows:

{|{}, {a, b, c, ...}, ...|} > {|a, b, c, ...|} where no number of members in both sets is accessible to the platonic objective level of existence of these sets.

Actually the platonic level is a common objective existence for both non-platonic subjective level of existence of members, where only at this level a contradiction like J membership, enables to conclude that J existence is false (which has an influence on P(A) and A identity, but not on P(A) and A platonic level of common objective existence).

Again, the platonic objective level of set is a discovered tautology, where the non-platonic subjective level of members is an invetion that is not necessarily tautology (as can be seen in the case of J).
 
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doronshadmi said:
Let's use Cantor's Theorem in order to explicitly demonstrated the difference between the platonic level of existence and the non-platonic level of existence.
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Don't bother. Cantor's Theorem is forged in Mathematics--rigorous, consistent, and well-defined. Shadmized philosophy, not so much. Whether you accept or agree with Cantor on a philosophic basis has no impact on the math.

...
2. There is at least one member of P(A) (let's call it S), which is defined as "the set of all A members, which are not members of the P(A) members that are paired with them"
...
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And this shows another reason you shouldn't be commenting on Cantor's Theorem. You cannot even express the proof correctly. Much like your mindless repetition of the Axiom of Infinity, you omit rather important detail and you insert things that simply aren't there. And you get things ass-backwards.
 
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doronshadmi said:
Let's use Cantor's Theorem in order to explicitly demonstrated the difference between the platonic level of existence and the non-platonic level of existence.

<ERRONEOUS TRAIN OF THOUGHT CUT OUT>

Again, the platonic objective level of set is a discovered tautology, where the non-platonic subjective level of members is an invetion that is not necessarily tautology (as can be seen in the case of J).
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And again, the above only is of the category 'observation' (for want of a better word).

Doron never reaches anything near the category 'conclusion'.

If one were to take the whole of the 7+ year thread of Doron's meandering philosophizing, one could only conclude that Organic Mathematics never has lifted off (one may even be so bold as to conclude it never reaches the launchpad...).

Does anyone see anything of worth in this current strain of Doronetics?
 
jsfisher said:
you omit rather important detail and you insert things that simply aren't there.
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Please demonstrate your claim in details, according to what I wrote in here
...
2. There is at least one member of P(A) (let's call it S), which is defined as "the set of all A members, which are not members of the P(A) members that are paired with them"
...
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Here are two examples of such P(A) member, that is not paired with any A member:



example 1:

{} (which is some P(A) member) is defined as "the set of all A members, which are not members of the P(A) members that are paired with them", by the following example:
Code: 
{ a , b , c , ... }
  ↕   ↕   ↕    
{{a},{b},{c}, ... }
Since all A members are members of the P(A) members that are paired with them, we get {} (the empty set) as the P(A) member that is not paired with any A member.



example 2:

{a,b,c,...} (which is some P(A) member) is defined as "the set of all A members, which are not members of the P(A) members that are paired with them", by the following example:
Code: 
{ a , b , c , ... }
  ↕   ↕   ↕    
{{b},{c},{d}, ... }
Since all A members are not members of the P(A) members that are paired with them, we get {a, b, c, ...} (set A) as the P(A) member that is not paired with any A member.


I am waiting to your detailed demonstration of your quoted claim above.
 
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doronshadmi said:
jsfisher said:
you omit rather important detail and you insert things that simply aren't there.
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Please demonstrate your claim in details, according to what I wrote in here
...
2. There is at least one member of P(A) (let's call it S), which is defined as "the set of all A members, which are not members of the P(A) members that are paired with them"
...
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  • Your set A is a specific set, yet Cantor's Theorem doesn't deal with any specific set.
  • "members that are paired" -- what pairing would that be?
  • "the set of all A members" -- an odd bit of gibberish. Did you mean "the set of all the members of A"? Wouldn't that be just the set A? Or maybe you haven't mastered the fine art of the comma.
  • "(let's call it S)" -- Why? You don't mention S again.
  • Where in the proof of Cantor's Theorem is "at least one member of P(A)" defined as you claim? (Order is important. A constructed set that is then shown to be a member of P(A) wouldn't be the same thing.)
  • ...
 
jsfisher said:
*Your set A is a specific set, yet Cantor's Theorem doesn't deal with any specific set.
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{a,b,c, ...} is an example without loss of generality of set A, where {{}, {a, b, c, ...}, ...} is an example without loss of generality of set P(A) (the power set of A), so we no not deal with any specific set or its power set.

jsfisher said:
* "members that are paired" -- what pairing would that be?
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See the examples (which are without loss of generality) in http://www.internationalskeptics.com/forums/showpost.php?p=10030370&postcount=3795.

jsfisher said:
* Or maybe you haven't mastered the fine art of the comma.
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I agree with your remark, thank you. At has to be "the set of all A members that are not members of the P(A) members that are paired with them".

jsfisher said:
* "(let's call it S)" -- Why? You don't mention S again.
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S is used here without a loss of generality as "the set of all A members that are not members of the P(A) members that are paired with them", and it is mentioned more than once in http://www.internationalskeptics.com/forums/showpost.php?p=10029127&postcount=3792.

jsfisher said:
* Where in the proof of Cantor's Theorem is "at least one member of P(A)" defined as you claim?
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By using Cantor's theorem please conclude that |P(A)|>|A| without "at least one member of P(A) that is not paired with any member of A".

jsfisher said:
*...
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*...

So jsfisher, thank you for your remark about my (,) wrong use, but since it was corrected it can be seen that you actually have no case (according to your attitude, papers like
http://www.math.umaine.edu/~farlow/sec26.pdf
http://www.math.ucla.edu/~hbe/resource/general/131a.3.06w/cantor.pdf
should not be written, so?).
 
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doronshadmi said:
{a,b,c, ...} is an example without loss of generality of set A, where {{}, {a, b, c, ...}, ...} is an example without loss of generality of set P(A) (the power set of A), so we no not deal with any specific set or its power set.
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So, you agree this claim of "without loss of generality" was something you left out of the current presentation.

See the examples (which are without loss of generality) in http://www.internationalskeptics.com/forums/showpost.php?p=10030370&postcount=3795.
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So, you agree these examples were something you left out of the current presentation.

I agree with your remark, thank you. At has to be "the set of all A members that are not members of the P(A) members that are paired with them".
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Now that you have addressed the extraneous comma issue, perhaps you'd consider dealing with the gibberish aspect.

S is used here without a loss of generality as "the set of all A members that are not members of the P(A) members that are paired with them", and it is mentioned more than once in http://www.internationalskeptics.com/forums/showpost.php?p=10029127&postcount=3792.
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So, you agree this other mention was something you left out of the current presentation.

By using Cantor's theorem please conclude that |P(A)|>|A| without "at least one member of P(A) that is not paired with any member of A".
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Since Cantor's Theorem establishes that |P(A)| is strictly greater than |A|, there is nothing further needed to conclude |P(A)| > |A|.

However, this is an aside and unrelated to the point you were (mis-)responding to.
 
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jsfisher said:
So, you agree this claim of "without loss of generality" was something you left out of the current presentation.
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I agree with you that I wrongly assumed that your professional level of mathematics enables you to understand that {a,b,c,...} or {{},{a,b,c,...},...} are expressions that are used "without loss of generality".

Now I realize that my assumption was wrong.

jsfisher said:
Since Cantor's Theorem establishes that |P(A)| is strictly greater than |A|, there is nothing further needed to conclude |P(A)| > |A|.
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Cantor's Theorem establishes that |P(A)| is strictly greater than |A|, exactly because it establishes that there is P(A) member that is not paired with any A member.
 
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