Cont: Deeper than primes - Continuation 2

[doronshadmi]

Set theory cannot prove a false theorem.

You are still avoiding an explicit demonstration of how ZF(C) set theory proves that the statement "the matrix includes all of the proper infinite subsets of the set N" is a false statement, or in other words, it able to disprove(= prove it false) it.

Moreover, there is no such thing like false theorem, because a given statement is considered as a theorem only if it proved to be True, under a given mathematical theory.

 

[jsfisher]

You are still avoiding an explicit demonstration of how ZF(C) set theory [dis-]proves that the statement...

Not my job. You made the baseless claim; the burden to prove it or disprove it falls to you.

No one here is responsible for disproving every bit of nonsense you assert as your next obvious truth.

 

[doronshadmi]

Not my job. You made the baseless claim

Since you claim that ZF(C) is The foundation of Mathematics, then this is your job to disprove the statement "the square matrix includes all of the proper infinite subsets of the set N".

If you can't do that, it is clearly shown that your claim that ZF(C) set theory is The foundation of Mathematics, does not hold water.

 

[doronshadmi]

No one here is responsible for disproving every bit of nonsense you assert as your next obvious truth.

I do not assert that the statement "the square matrix includes all of the proper infinite subsets of the set N" is an obvious truth.

All I ask from you is to prove OR disprove it by ZF(C) set theory, exactly because you are the one who claims that ZF(C) set theory is The foundation of Mathematics.

If, this time, you are unable to prove OR disprove it by ZF(C) set theory, then your claim that ZF(C) set theory is The foundation of Mathematics, does not hold water.

 

[doronshadmi]

jsfisher, in order to be clearer, by using ZF(C) please prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

You did not do it yet, and I suspect that you can't do it.

 

[Little 10 Toes]

doronshadmi, please show the exact post where jsfisher states that "ZF(C) is the foundation of mathematics".

 

[jsfisher]

doronshadmi, please show the exact post where jsfisher states that "ZF(C) is the foundation of mathematics".

It is best ignored. At this point, doronshadmi is following his typical pattern of asserting utter nonsense, modifying his nonsense under fire, defending his nonsense with convolutions and continual goalpost shifts, shifting the burden, and then finally running off to lick wounds. Rinse, lather, repeat.

We are at the penultimate step. The final step could be long or it could be short....we'll see.

 

[doronshadmi]

doronshadmi, please show the exact post where jsfisher states that "ZF(C) is the foundation of mathematics".

All is needed is to see jsfisher's response at the end of http://www.internationalskeptics.com/forums/showpost.php?p=10998703&postcount=939 and also his response in http://www.internationalskeptics.com/forums/showpost.php?p=10999186&postcount=941, in order to know what is ZF(C) for him.

Now, by using his favorite theory (which is ZF(C)), jsfisher is politely asked to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

 

[doronshadmi]

defending his nonsense ...

I do no defend anything.

All I ask from you, jsfisher, is this:

Please use your favorite theory (which is ZF(C) according to

I've always favored ZFC (although the C part does cause me some discomfort) ...

) in order to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

So please defend on your favorite theory (which is ZF(C)), by prove (by using it) that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

 

[Little 10 Toes]

Since you claim that ZF(C) is The foundation of Mathematics, then this is your job to disprove the statement "the square matrix includes all of the proper infinite subsets of the set N".

If you can't do that, it is clearly shown that your claim that ZF(C) set theory is The foundation of Mathematics, does not hold water.

I do not assert that the statement "the square matrix includes all of the proper infinite subsets of the set N" is an obvious truth.

All I ask from you is to prove OR disprove it by ZF(C) set theory, exactly because you are the one who claims that ZF(C) set theory is The foundation of Mathematics.

If, this time, you are unable to prove OR disprove it by ZF(C) set theory, then your claim that ZF(C) set theory is The foundation of Mathematics, does not hold water.

doronshadmi, please show the exact post where jsfisher states that "ZF(C) is the foundation of mathematics".

All is needed is to see jsfisher's response at the end of http://www.internationalskeptics.com/forums/showpost.php?p=10998703&postcount=939 and also his response in http://www.internationalskeptics.com/forums/showpost.php?p=10999186&postcount=941, in order to know what is ZF(C) for him.

Now, by using his favorite theory (which is ZF(C)), jsfisher is politely asked to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

So you lied when you said "... you [jsfisher]claim that ZF(C) is The foundation of Mathematics..." and " ... your claim that ZF(C) set theory is The foundation of Mathematics, does not hold water" in the first quoted message.

So you lied when you said " ... exactly because you [jsfisher] are the one who claims that ZF(C) set theory is The foundation of Mathematics" and "... your claim that ZF(C) set theory is The foundation of Mathematics, does not hold water" in your second quoted message.

You can't even say, "Oops, sorry. You didn't say those things." You have to say, "Well, read these two messages." Neither one of those messages have what you claim that jsfisher said. The closest thing is a phrase that came from a paper that, if I remember correctly, *you* introduced. That phrase? "Classical Cantorian set theory as developed over the last one hundred years, and as formalized in its most popular form in Zermelo-Fraenkel set theory, is still our best candidate for a secure foundataion [sic] for mathematics."

 

[doronshadmi]

So you lied when you said ...

Not at all.

ZF(C) is clearly jsfisher's favorite theory by his own words:

I've always favored ZFC (although the C part does cause me some discomfort) ...

So, by using his favorite theory, jsfisher is asked to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

 

[jsfisher]

ZF(C) is clearly jsfisher's favorite theory by his own words:

I've always favored ZFC (although the C part does cause me some discomfort) ...

Ignoring the discrepancy between 'favorite' and 'favored', I find the ellipses curious. The quotation continued as follows:

...but you are the one with the assertion. How about you tell us how you intend to prove that the cardinality of the set of all infinite subsets of N has the same cardinality as N.

It is useful, too, to look at the question that led to the quoted text:

So what axiomatic formal system of sets has to be used in order to prove or disprove it?

Do you have any suggestions?

Nowhere near what Doronshadmi claimed, so, yes, he lied.



As I have previously stated, we are at the penultimate step in the cycle where Doronshadmi tries to shift the burden. Rinse, lather, repeat.

 

[doronshadmi]

As I have previously stated, we are at the penultimate step in the cycle where Doronshadmi tries to shift the burden. Rinse, lather, repeat.

Ok jsfisher, we are at the step that, by using your favored theory (which is ZF(C)), please prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

No assertion of any kind is involved in my request to you.

 

[doronshadmi]

Nowhere near what Doronshadmi claimed, so, yes, he lied.

Here it is:

So what axiomatic formal system of sets has to be used in order to prove or disprove it?

Do you have any suggestions?

I've always favored ZFC (although the C part does cause me some discomfort) ...

Since you have always favored ZFC (although the C part does cause you some discomfort), then please use it in order to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

 

[doronshadmi]

At this point, doronshadmi is following his typical pattern of asserting utter nonsense, modifying his nonsense under fire, defending his nonsense with convolutions and continual goalpost shifts, shifting the burden, and then finally running off to lick wounds. Rinse, lather, repeat.

Well after you showed to the posters here what a "great" moderator you are, please show to the posters here what a great mathematician you are, by using ZF(C) (your favored theory) in order to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

 

[doronshadmi]

... to shift the burden ...

Well this is ridiculous.

Real mathematicians do not avoid challenges, on the contrary they are doing their best in order to prove things, and in this case you are asked to deal with the following challenge:

Please use your favored theory (which is ZF(C), in this case) in order to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

Maybe http://www.internationalskeptics.com/forums/showpost.php?p=11002986&postcount=951 leads to such proof if the ur-elements there (which are used in order to prove that |{1,1,1,...,2,2,2,...,3,3,3,...,...}| = |{1,2,3,...}| = |N|) can be translated into pure sets.

Also I wish to correct some mistake of mine (written at the end of http://www.internationalskeptics.com/forums/showpost.php?p=10999532&postcount=948), here it is:

So, the matrix has |N| objects and so is the case about the diagonal mutiset {1,1,1,...,2,2,2,...,3,3,3,...,...} that does not miss any member in the set of all proper infinite subsets of set N.

I can't claim that the diagonal mutiset {1,1,1,...,2,2,2,...,3,3,3,...,...} does not miss any member in the set of all proper infinite subsets of set N without a rigorous proof.

Therefore I'v asked jsfisher to use his favored theory (which is ZF(C), in this case) in order to prove that the cardinality of the set of all proper infinite subsets of set N is strictly greater than |N|.

By doing that he actually proves that the set of all natural numbers (notated as N), is not the one and only one infinite set of natural numbers that is not in the range of |N| proper infinite subsets of set N.

If it can't be proven within ZF(C) it means that ZF(C) is not sufficient in order to be considered as the foundation of mathematics.

 

[jsfisher]

... to shift the burden ...

Well this is ridiculous.
...
I can't claim...without a rigorous proof.

Therefore I'v asked jsfisher...to prove....

"...to shift the burden...". Apparently not so ridiculous.

 

[Little 10 Toes]

So you lied when you said ...

Not at all.

ZF(C) is clearly jsfisher's favorite theory by his own words:

I'm going to stop you right there. I'm not even going to quote any other portion of your post.

Not only do you tell lies, you don't read posts. In fact, you did not read my post.

I showed you what was truly said and specifically what you said. I even gave you an opportunity to fix it up, but you did't take it. In fact, you tried to change what you said. Too bad the posts show what you have done.

I have lost all respect for you. You have shown your true colors one too many times.

I don't care what you say next. It really doesn't matter.

Much respect to jsfisher for putting up with you and your actions.

 

[doronshadmi]

"...to shift the burden...". Apparently not so ridiculous.

Again, real mathematician actually welcomes mathematical challenges.

Since "...to shift the burden..." is your response to mathematical challenges, it seems that you do not have the needed passion in order to be considered as real mathematician.

I am sure that you know a lot about the current discussed subject, and one of the important things that I have learned from you about the current subject, is the difficulty to use diagonalization among non-ordered collections.

ZF(C) is considered as a pure-set theory where order is insignificant, which means that diagonalization can't be used directly within ZF(C), or in other words, some technique has to be used in order to use ordered things, such that they are based on non-ordered things.

For example, the representation of ordered pairs in terms of pure sets (where order is insignificant among pure sets) (for example: Kuratowski definition as seen in https://en.wikipedia.org/wiki/Ordered_pair).

Since you can't or don't wish to be a participator of such challenge, I have no choice but to do it by myself, but basically I think that mathematical development has to be done not only individually.

------------------

Any way since this is the current state (you can't or don't wish to be a participator of such challenge) I'll try to do it by myself as best I can.

Here it is:

Each member in the set of proper infinite subsets of set N is mapped with its disjoint set of N members.

If I am not wrong, this set of all disjoint sets is actually the power set of set N (notated as P(N)).

So, if I am not wrong, the cardinality of the set of all proper infinite subsets of set N can't be less than |P(N)|.

----------------------

Now, let's return to Cantor's proof of Cantor's theorem:

1) It is impossible to determine the set of all P(N) members, such that no one of these unique members is mapped with some unique member of set N.

2) Order is insignificant within ZF(C) (which is a pure set theory), so I don't see how diagonalization can directly be used among sets, in order to prove that |P(N)| > |N| (and do not forget that we are dealing here with infinite sets).

3) By not ignoring (1) and (2), I still do not see how exactly Cantor proved his theorem in case of infinite sets.

In that stage, jsfisher, I need your help on order to clearly show that Cantor's proof of his theorem, actually uses diagonalization among sets (where order is insignificant and sets are pure), such that he actually proves that it is not closed under the equation |N|+1=|N|.

 

[doronshadmi]

I showed you what was truly said and specifically what you said.

Wrong Little 10 Toes, you simply have missed jsfisher's attitude about ZF(C) as his favored theory of the foundation of mathematics, as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=11029453&postcount=1008.

----------------

I am not surprised that you have missed jsfisher's attitude about ZF(C).

After all, after hundreds of posts about cardinality of infinite sets (when you are one of the active participators of these posts) a post like http://www.internationalskeptics.com/forums/showpost.php?p=10952509&postcount=811 clearly demonstrates that you are not doing your best in order to understand a given subject, before you reply about it.