Cont: Deeper than primes - Continuation 2

[doronshadmi]

"By using Cantor's theorem" we don't need to consider any mappings between S and P(S). Cantor's theorem tells us |S| < |P(S)| directly.

Please explain how exactly "Cantor's theorem tells us |S| < |P(S)| directly" without the need "to consider any mappings between S and P(S)"?

 

[jsfisher]

Please explain how exactly "Cantor's theorem tells us |S| < |P(S)| directly" without the need "to consider any mappings between S and P(S)"?

Cantor's Theorem: The cardinality of any set is less than the cardinality of its power set.

That's Cantor's Theorem. That's all there is; nothing more. No mappings whatsoever.

 

[doronshadmi]

Cantor's Theorem: The cardinality of any set is less than the cardinality of its power set.

That's Cantor's Theorem. That's all there is; nothing more. No mappings whatsoever.

jsfisher, a theorem is a statement that has been proven (and in this case the statement is: "The cardinality of any set is less than the cardinality of its power set.").

So please explain exactly how Cantor's statement has been proven with "No mappings whatsoever."

In other words, how exactly it is possible "to show no bijective mapping exists for infinite sets" by "No mappings whatsoever."?

Moreover, isn't f is the notation for function (or mapping) as a part of Cantor's Theorem?

 

[jsfisher]

jsfisher, a theorem is a statement that has been proven (and in this case the statement is: "The cardinality of any set is less than the cardinality of its power set.").

So please explain exactly how Cantor's statement has been proven with "No mappings whatsoever."

I never said it could. The context was using Cantor's Theorem, not proving it. It has been proven; we don't need to do that again and again.

The proof of the theorem is not part of the theorem. Cantor's Theorem is the mathematical fact that |S| < |P(S)|. Done.

In other words, how exactly it is possible "to show no bijective mapping exists for infinite sets" by "No mappings whatsoever."?

If the goal is to show two arbitrary sets are of different cardinality, than you'd need to show no bijective mapping between the two were possible. Cantor's Theorem would not apply.

On the other hand, if one of the sets happened to be the power set of the other, then citing Cantor's Theorem is sufficient. Done.

And finally, if your goal was to prove Cantor's Theorem (as if that were necessary), then the proof used by Cantor is readily available.

Moreover, isn't f is the notation for function (or mapping) as a part of Cantor's Theorem?

No. The proof of a theorem is not part of a theorem.

 

[doronshadmi]

No. The proof of a theorem is not part of a theorem.

A theorem is a statement that has been proven so, a theorem is not less than a statement AND its proof, otherwise it can't be considered as a theorem.

I have searched "The proof of a theorem is not part of a theorem" by google, and this is what I have got https://www.google.co.il/?gfe_rd=cr...+proof+of+a+theorem+is+not+part+of+a+theorem" .

So please support your claim by some link to some professional mathematical material.

 

[jsfisher]

A theorem is a statement that has been proven so, a theorem is not less than a statement AND its proof, otherwise it can't be considered as a theorem.

I have searched "The proof of a theorem is not part of a theorem" by google, and this is what I have got https://www.google.co.il/?gfe_rd=cr...+proof+of+a+theorem+is+not+part+of+a+theorem" .

So please support your claim by some link to some professional mathematical material.

Doronshadmi, this is absurd.

Is it really your contention that to apply the Pythagorean Theorem, for example, to show a 4 x 3 rectangle has 5 for its diagonal, I must revisit the proof for the theorem?

I'll bet you are the life of the party every time the Intermediate Value Theorem comes into play. And let's not forget the Four-color Theorem which involves analysis of 1,936 special cases by computer.


ETA: And what about theorems that have multiple proofs? (That would be all of them, by the way.) Must all possible proofs be brought into consideration?

 

[doronshadmi]

I'll bet you are the life of the party every time the Intermediate Value Theorem comes into play.

Also in this case functions (mappings) are needed, and they are also parts of the formal proof of the Intermediate Value Theorem.

Cantor's Theorem is the mathematical fact that |S| < |P(S)|. Done.

This is not a mathematical fact without functions (mappings), and in the case of Cantor's Theorem they are inseparable of a given indirect proof (and in this case, proof by contradiction), so your "No mappings whatsoever" is not true in the case of Cantor's theorem, and this is actually my argument.

And what about theorems that have multiple proofs? (That would be all of them, by the way.) Must all possible proofs be brought into consideration?

No. But if a theorem is not the Intermediate Value Theorem, it is not less than a given statement AND a given proof, and in the case of Cantor's Theorem it is inseparable of a given indirect proof (and in this case, proof by contradiction), so your "No mappings whatsoever" is not true in the case of Cantor's theorem, and this is actually my argument.

Is it really your contention that to apply the Pythagorean Theorem, for example, to show a 4 x 3 rectangle has 5 for its diagonal, I must revisit the proof for the theorem?

No, but if you wish the understand The Pythagorean Theorem more generally, I thing it is a good idea to revisit at least one of its multiple proofs.

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

Now, I wish to discuss about Cantor's theorem, which in this case is not less than a given statement AND a given indirect proof (and in this case, proof by contradiction), where functions (mappings) are inseparable parts of that proof (again, your "No mappings whatsoever" is not true in the case of Cantor's theorem, and in order to be clearer this is actually my argument).

So please let's be focused only on Cantor's theorem in our next replies.

 

[doronshadmi]

For finite sets, the existence of an injective mapping that isn't bijective is sufficient to show that no bijective mapping exists.

... the existence of an injective mapping that is not bijective is not sufficient to show no bijective mapping exists for infinite sets.

So you distinguish between the sufficiency of mappings in case of finite or infinite sets.

I agree with you, because one of the reasons of this distinction is that if, for example, S is a finite set, then |S| ≠ |S|+1, where in case that S is an infinite set |S| = |S|+1.

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

Now I wish to write something about Cantor's Theorem, as follows:

If S is a finite set, for example, S={a,b,c}, it is clear that "the existence of an injective mapping that isn't bijective is sufficient to show that no bijective mapping exists" between S and its power set {{},{a},(b},{c},{a,b},{a,c},{b,c},{a,b,c}}.

For example, by the particular case

a --> {a}
b --> {b}
c --> {c}

we get (by the proof of Cantor's statement) {} that is not mapped with any member of S, and by this particular case we can use |{a,b,c}| < |{{},{a},(b},{c}}| in order to conclude that |{a,b,c}| < |{{},{a},(b},{c},{a,b},{a,c},{b,c},{a,b,c}}|.

My argument is that this one particular case "is sufficient to show that no bijective mapping exists", because in case that S is finite, |S| ≠ |S|+1 .

If S is an infinite set, for example, S={a,b,c,...}, then by

a --> {a}
b --> {b}
c --> {c}
...

we also get (by the proof of Cantor's statement) {} that is not mapped with any member of S, but unlike in the case of finite sets, "the existence of one injective mapping that is not bijective, is not sufficient to show no bijective mapping exists for infinite sets", because in case that S is infinite, |S| = |S|+1 .

Please carefully pay attention to the following argument about the proof of Cantor's statement:

For |P(S)| mappings between S and P(S) (where S is infinite) any given mapping is between all |S| members of S and |S| members of P(S), which defines exactly one member of P(S) that is not mapped with any one of all S members.

So for |P(S)| mappings between S and P(S) it is true that Cantor's proof is only about |S| and |S|+1 members for each mapping, but since S is infinite |S| = |S|+1, so no |P(S)| mappings (where each one of them is only about |S| = |S|+1) are "sufficient to show that" |S| < |P(S)|, in case that S is an infinite set.

 

[jsfisher]

Doronshadmi, this is still absurd.

Is it really your contention that to apply the Pythagorean Theorem, for example, to show a 4 x 3 rectangle has 5 for its diagonal, I must revisit the proof for the theorem?

 

[doronshadmi]

Doronshadmi, this is still absurd.

Is it really your contention that to apply the Pythagorean Theorem, for example, to show a 4 x 3 rectangle has 5 for its diagonal, I must revisit the proof for the theorem?

Again, the answer is no, but if you wish to understand better The Pythagorean Theorem, I think it is a good idea to revisit at least one of its multiple proofs.

Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728.

Thank you.

 

[jsfisher]

Again, the answer is no, but if you wish to understand better The Pythagorean Theorem, I think it is a good idea to revisit at least one of its multiple proofs.

So, we can use the Pythagorean Theorem alone and accept that it is proven as a given. Why is Cantor's Theorem different?

By the way, are you accepting |Q| = |Q|+1 for non-finite set Q as some sort of proven theorem? You need to provide the proof since theorem's and proofs are inseparable (except in Pythagoras' case, of course). That will, of course, get into transfinite numbers and all sorts of more theorems (and their inseparable proofs) all of which have Cantor's Theorem at their foundation.

You also have referred to a function f() as part of Cantor's Theorem / proof. How is f() defined, exactly?

 

[doronshadmi]

jsfisher, please reply to the content of http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728 in details.

Thank you.

 

[jsfisher]

jsfisher, please reply to the content of http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728 in details.

Thank you.

The issues raised in http://www.internationalskeptics.com/forums/showpost.php?p=10944808&postcount=731 are all relevant. Please reply to its content in detail.

Thank you.

 

[doronshadmi]

The issues raised in http://www.internationalskeptics.com/forums/showpost.php?p=10944808&postcount=731 are all relevant.

Great. In that case please relate them to the content of http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728 in order to clearly show their relevance to what is written in this post.

 

[jsfisher]

Great. In that case please relate them to the content of http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728 in order to clearly show their relevance to what is written in this post.

No. You may either respond to my post or not; that is your choice. I have the same option with respect to yours.

 

[doronshadmi]

No. You may either respond to my post or not; that is your choice. I have the same option with respect to yours.

Why is Cantor's Theorem different?

Because according to what is written in http://www.internationalskeptics.com/forums/showpost.php?p=10944808&postcount=731 it is not a theorem, but it is a conjecture.

To show otherwise, please show the mistakes in http://www.internationalskeptics.com/forums/showpost.php?p=10944808&postcount=731.

 

[jsfisher]

Because according to what is written in http://www.internationalskeptics.com/forums/showpost.php?p=10944808&postcount=731 it is not a theorem, but it is a conjecture.

To show otherwise, please show the mistakes in http://www.internationalskeptics.com/forums/showpost.php?p=10944808&postcount=731.

I made no mistakes in that post, nor does that post in any way suggest that Cantor's Theorem is unproven.

If, on the other hand, you'd like to contest the standard proof of Cantor's Theorem, you are free to do so. However, it would be best if you began by quoting word-for-word the standard proof with added commentary, clearly separated from the actual proof text, that points out any error in the proof.

 

[doronshadmi]

I made no mistakes in that post, nor does that post in any way suggest that Cantor's Theorem is unproven.

I have used the wrong link, the right one is http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728.

So here it is again (with the right link).

No. You may either respond to my post or not; that is your choice. I have the same option with respect to yours.

Why is Cantor's Theorem different?

Because according to what is written in http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728 it is not a theorem, but it is a conjecture.

To show otherwise, please show the mistakes in http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728.

 

[jsfisher]

I have used the wrong link, the right one is http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728.

So here it is again (with the right link).




Because according to what is written in http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728 it is not a theorem, but it is a conjecture.

To show otherwise, please show the mistakes in http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728.

You skipped over this part:

If, on the other hand, you'd like to contest the standard proof of Cantor's Theorem, you are free to do so. However, it would be best if you began by quoting word-for-word the standard proof with added commentary, clearly separated from the actual proof text, that points out any error in the proof.

 

[doronshadmi]

You skipped over this part:

If, on the other hand, you'd like to contest the standard proof of Cantor's Theorem, you are free to do so. However, it would be best if you began by quoting word-for-word the standard proof with added commentary, clearly separated from the actual proof text, that points out any error in the proof.

Thank you for your advice of how to properly write an article about such subject to some professional mathematical journal, but this is not the case here.

You skipped over this post:

http://www.internationalskeptics.com/forums/showpost.php?p=10944628&postcount=728.

If you skip over it once again, your evasion will be clearly noted about my argument.