Re: Contradicting Cantor/Counting the Power set.  sci.math 21 Nov 1998 


In article  Mike Deeth, [email protected] writes:

>However, it turns
>out that the power set is indeed countable, as may be seen
>by examining the following table:
>
[... ...         ...                   ...]
>
[... ...         ...                   ...]
>
>Nathaniel Deeth
>Age 11
>(http://www.mathacademy.com/platonic_realms/minitext/infinity.html)

Hi Deeth, I asked exactly the same question to my tutor, about seven
years ago, showing to him a beautiful table likewise the one given by you.
I told him it was really evident that all the subsets of the naturals numbers 
could be generated, so that the  assertion of uncountability of the set of all the
subsets should have to be wrong, as it is well-know that a generating algorithm is 
equivalent to a function of enumeration.   For my astonishment his answer was 
that my table was only a sort of construction of the ordinals. Into my table I had
simply re-constructed the ordinals. Now I am sure he was,  he is still today, a 
smart teacher, but at that time his answer was very displeasing for me.

How was it possible? I could effectively count the set of "all" the
subset (of naturals or not, as you can substitute any other set of objects
{a,b,c,d,....} to {1,2,3,4,....} into your table) and in the mean-while
the set was uncountable ! and when I founded an arrangement, a list, 
a combination, an order then naturals were no longer cardinals but ordinals...??
Naturals had such chameleonic nature to be cardinals when were uncountable and
ordinals when were countable.

This fact to be in the mean-while countable and uncountable was not the
only one example; for the plain satisfaction of my curiosity I began to list
all the similar cases within mathematics:
-the Skolem paradox,
-the Lowenheim-Skolem theorem,
-the fact that in set theory the axiom of choice is equivalent to 
the Zorn Lemma equivalent to the well-ordering theorem 
((where is the well-order for reals?)) 
equivalent to the principle of enumeration, and that this latter 
states " EACH SET CAN BE ENUMERATED "
quiet together with the assertion of existence of UNCOUNTABLE sets,
-... (maybe other cases I forget now) ...

In logic "to be countable" is a property exactly like "to be round" or 
"to be smiling", but when " the round ball is not round" we get a situation 
like "A and not A", 
which is a contradiction, a failure, a hole ...

Mathematics began to appear to me like a type of cheese (I don't know
if you know it) called Emmental with many holes.
......

Paola
Age 10
(http://www.floweracademy/bologna_realms/wanderingful_town/decidabletext.html)
 





Re: Contradicting Cantor/Counting the Power set.  sci.math 23 Nov 1998 


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>Xref: leporello.cs.unibo.it sci.math:223914
In article  Virgil Hancher, [email protected] writes:

>In article <[email protected]>, pc
> wrote:
>
>> all the similar cases within mathematics:
>> -the Skolem paradox,
>> -the Lowenheim-Skolem theorem,
>> -the fact that in set theory the axiom of choice is equivalent to the Zorn
>> Lemma equivalent to the well-ordering theorem ((where is the well-order
>> for reals?)) 
>> equivalent to the principle of enumeration, and that this latter 
>> states " EACH SET CAN BE ENUMERATED "
>
>Enumerating and well-ordering are not the same.

OF COURSE.

>
>A set has been enumerated when it has been put into one-to-one
>correspondence with  a subset the natural numbers ( possibly the whole
>set).
>
>A set has been well-ordered when its elements have been ordered in such a
>way that any non-empty subset has a smallest element in this ordering.
>This well-ordering may be unrelated to any natural ordering of the set.
>
>A well-ordering of the power set of the natural numbers
>is NOT an enumeration of the power set.

Why not?


AC = Axiom of Choice
ZL = Zorn s Lemma
WO = Well-Ordering Principle
PE = Principle of Enumeration
TR = Tricotomy (x)(y)(x=  ZL  <->  WO   <->  PE   <->  TR ]


References

Model Theory, C.C. Chang, H.J. Keisler, North-Holland 1973 (see Appendix A)
Introduction to Mathematical Logic, Elliott Mendelson, Wadsworth & Brooks, 1987
(see page  213)

I strongly suggest also

Beyond First-order Logic: The Historical Interplay between Mathematical
Logic and Axiomatic Set Theory, Gregory H. Moore, in 
History and Philosophy of Logic, 1 (1980), 95-137