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Topic: Re: When has countability been separted from listability?
Replies: 5   Last Post: Oct 3, 2017 5:22 PM

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Re: When has countability been separted from listability?
Posted: Oct 3, 2017 3:22 PM
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On Sunday, October 1, 2017 at 10:11:25 PM UTC-7, WM wrote:
> Am Montag, 2. Oktober 2017 00:39:51 UTC+2 schrieb John Dawkins:
> > In article <fb18e185-529b-4288-92d8-18268a0912cc@googlegroups.com>,
> > WM <wolfgang.mueckenheim
> >

> > > Cantor has shown that the rational numbers are countable by constructing a
> > > sequence or list where all rational numbers appear. Dedekind has shown that
> > > the algebraic numbers are countable by constructing a sequence or list where
> > > all algebraic numbers appear. There was consens that countability and
> > > listability are synonymous. This can also be seen from Cantor's collected
> > > works (p. 154) and his correspondence with Dedekind (1882).
> > >
> > > Meanwhile it has turned out that the set of all constructible real numbers is
> > > countable but not listable because then the diagonalization would produce
> > > another constructible but not listed real number.
> > >
> > > My questions:
> > > (1) Who realized first that countability is not same as listability?
> > > (2) Who has decided that this is not contradiction in set theory?

> >
> > Define "listable".

>
> "Consider any point set M which [...] has the property of being countable such that the points of M can be imagined in the form of a sequence". [Cantor, collected works, p. 154] "[...] ordering of all algebraic numbers in a sequence, their countability". [G. Cantor, letter to R. Dedekind (10 Jan 1882)]
>
> Regards, WM


Here "any point set M" in countable set theory
isn't for example "all the points in space"
(that is effective for all kinds of things and
many effective terms). Then for R or the Real
Zahlen for example as "any point set M for example R"
then for Cantor from the context that's a sequence.

"Cantor proves the line is drawn."

That's "countable".




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