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Topic: § 534 Finis
Replies: 61   Last Post: Aug 13, 2014 3:18 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: § 534 Finis
Posted: Aug 6, 2014 11:46 AM

On Wednesday, 6 August 2014 16:24:42 UTC+2, Ben Bacarisse wrote:
> mueckenh@rz.fh-augsburg.de writes:
>
>
>

> > Set theorists claim that all rational numbers can be indexed by all
>
> > natural numbers. In § 533 I have shown not only that every natural
>
> > number n fails

> (as everyone would agree)

Not those who believe in indexings by natural numbers.
>

> (but there are much simpler ways to show the same thing

Of course. But not so convincing. It should be enough to show that there are indexings that with certainty are not bijections. Alas, if the brain is blinded ...

> -- you can
> construct a sequence of sets of "un-indexed" rationals that grows in
> almost any way yo like)

That should have raised doubts in Cantor.
>
>
>

> > Since nothing but
> > finite natural numbers are available for indexing, and provably al
> > fail, this task cannot be accomplished.

>

> Not using normal mathematics, no.

So we agree.

> The indexing is just a bijection and
> is simple to construct, as has been shown many times. What's more, you
> agree[1]:
>
>
>
> | Usually infinity mans actual infinity. If I use infinity (in my book)
> | then it means potential infinity. A bijection means that for every
> | element of one set there is always one and only one of the other
> | set. Further it means that all elements that can be used will be used
>
>
>
> You give several examples of such infinite bijections in you book, and
> this function from N to Q+:

Of course. But that is obviously never a completed infinity.
>
>
> And I don't know what goes on in your mind because you refuse to answer

See above. Frequently answere. Potential infinity. No complete sets. No cardinality aleph_0.

>
> No, that much is clear, but you could answer simple questions about what
> is and is not true in your mathematics.

I did: with increasing n the number of unit intervals of rationals without any rational indexed by a natural less than n increases without bound, i.e., infinitely. Since nothing but finite natural numbers are available for indexing, and provably all fail, this task cannot be accomplished.

Regards, WM