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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: 5,511 Registered: 9/25/16
Re: When has countability been separted from listability?
Posted: Oct 3, 2017 3:42 PM

No clue that Cantor talked about algebraic numbers
and not about reals, in the cited passgage. Right?

Am Dienstag, 3. Oktober 2017 21:40:54 UTC+2 schrieb burs...@gmail.com:
> 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".

Date Subject Author
10/3/17 Guest
10/3/17 bursejan@gmail.com
10/3/17 bursejan@gmail.com
10/3/17 bursejan@gmail.com
10/3/17 ross.finlayson@gmail.com
10/3/17 bursejan@gmail.com