Re: induction vs Cantor

From: Poker Joker (Poker_at_wi.rr.com)
Date: 11/28/04 

Date: Sun, 28 Nov 2004 06:15:36 GMT

"Virgil" <ITSnetNOTcom#virgil@COMCAST.com> wrote in message
news:ITSnetNOTcom%23virgil-D65AFE.12191327112004@[63.218.45.211]...
> In article <5Q1qd.94258$T02.83660@twister.rdc-kc.rr.com>,
> "Poker Joker" <Poker@wi.rr.com> wrote:
>
>> "Virgil" <ITSnetNOTcom#virgil@COMCAST.com> wrote in message
>> news:ITSnetNOTcom%23virgil-DA19C7.22201926112004@[63.218.45.211]...
>>
>> <snipped>
>>
>> >> Cantor function from R^N to R? AFAIK, Cantor discussed a function
>> >> from
>> >> N to R.
>> >
>> > Cantor says that, given any such function from N to R, his construction
>> > produces a number not in image of the function.
>> >
>> > Thus that construction defines a function from the set of all functions
>> > from N to R, R^N, to R, which I have called the Cantor function.
>>
>> So you are claiming an uncountable number of fucntions without using
>> Cantor's conclusions. You are saying Cantor's work was a waste of
>> time because it's okay to just assume uncountability. It needs no proof.
>
> There is a different theorem that the power set is of larger cardinality
> than the set itself of any set. This other theorem does not use a an
> "anti-diagonal" construction. so I presumed that you had no objection to
> it. By this theorem, one can easily show that R^N is not countable. So
> no unproven assumptions about existence of uncountable sets is needed.

That's a crock of you-know-what. Your going to use the conclusion of
the proof because there's another way of proving it? You do what you want.

>> >> If you are going to start using Cantor's conclusions though, I am
>> >> going
>> >> to
>> >> have no further comment.
>> >
>> > That "Cantor function" is what the proof is all about. The steps of the
>> > proof show how to build the Cantor function in such a way that for each
>> > function, f:N -> R, Cantor(f) is not a member of Range(f).
>>
>> Yep, and in a proof of uncountable sets, one shouldn't utilize the fact
>> that
>> there are uncountable sets.
>
> The proof that uncountably infinite sets exist if countably infinite
> sets exist precedes and is independent of the proof that the reals are
> one of those uncountable sets.

You're still trying to utilize the conclusion that there are uncountably
many
reals.

Relevant Pages