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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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bursejan@gmail.com

Posts: 4,551
Registered: 9/25/16
Re: When has countability been separted from listability?
Posted: Oct 3, 2017 3:40 PM
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Fruit cake at its best...

Am Dienstag, 3. Oktober 2017 21:22:12 UTC+2 schrieb Ross A. Finlayson:
> 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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