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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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ross.finlayson@gmail.com

Posts: 2,700
Registered: 2/15/09
Re: When has countability been separted from listability?
Posted: Oct 3, 2017 4:04 PM
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On Tuesday, October 3, 2017 at 12:46:03 PM UTC-7, burs...@gmail.com wrote:
> Anyway fruit cake got salvaged, at least he
> wrote: Here "any point set M" in countable set theory
> isn't for example "all the points in space"
>
> But what is countable set theory? CZFC?
>
>
> Am Dienstag, 3. Oktober 2017 21:42:59 UTC+2 schrieb burs...@gmail.com:

> > 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".


What is countable set theory? It's just limits.
The continuum analysis in the countable set theory
is justly established as about the limits, then
for of course those being perfect.

The countably additivity for the character of real
analysis and measure 1.0 is for example all "countable"
the products of the copies of the "sets" of the
numbers or the ordinals. It's just the relation
of all the differentials in the effective model of
the continuous media.



And here's to fruitcake. The very idea of such
a conveyor of holiday cheer and goodwill toward
that the stored durable food with for example the
citrus and nuts in a good bread, often preservified
with the liqueur for the shelf stability, an enduring
food. Fruitcake or the fruitcake has a special place
in a holiday tradition , and any difference between
fruitcake and me is through no fault of fruitcake's,
here though I am not it. Let's all remember fruitcake
and good luck for you then if the year is so good there
is some fruitcake. And, when there isn't, sometimes it
suffices for it to be as good as it is even when there
isn't any.

It's the idea, the idea of fruitcake:
not just a food, a gift.

A treasure.

The idea of fruitcake is an enduring gift.

There's enough for everybody.






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