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Topic: When has countability been separted from listability?
Replies: 1   Last Post: Sep 30, 2017 1:18 PM

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Posts: 3,394
Registered: 10/18/08
Re: When has countability been separted from listability?
Posted: Sep 30, 2017 1:18 PM
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Am Samstag, 30. September 2017 17:59:07 UTC+2 schrieb John Gabriel:
> On Saturday, 30 September 2017 03:23:14 UTC-5, WM wrote:

> > My questions:
> > (1) Who realized first that countability is not same as listability?

> I am convinced that Cantor thought these mean exactly the same thing.

So it is. He wrote it only en passant because it was so clear:

Betrachten wir irgendeine Punktmenge (M), welche innerhalb eines n-dimensionalen stetig zusammenhängenden Gebietes A überalldicht verbreitet ist und die Eigenschaft der Abzählbarkeit besitzt, so daß die zu (M) gehörigen Punkte sich in der Reihenform

M1, M2, ..., M?,...

vorstellen lassen; (Collected works p. 154)

Vielleicht wird Sie eine Mittheilung interessiren, welche ich vor ein paar Tagen von Weierstraß erhalten habe.
Derselbe macht mich darauf aufmerksam, dass die von mir vor acht Jahren entdeckte Einordnung aller algebraischen Zahlen in Reihenform, ihre Abzählbarkeit, ein sehr fruchtbares und merkwürdiges Condensationsprincip von Singularitäten, (viel allgemeiner und einfacher als das Hankelsche, Ihnen wohl bekannte) abgiebt. (Cantor, letter to Dedekind, 1882)
> But it might make sense if defined the way I stated: A set is countable if its elements can be systematically named.

That is correct. But it fails for the constructible real numbers.
> Idiot professors have never had my depth of understanding so they simply instruct the students that if they can place a set in bijection with |N, then that set is countable. This works ONLY because the elements of |N can be systematically named.

Both is correct. But it does not work for the constructible real numbers.

> Of course orangutan morons (plenty on this forum) have never questioned why |N is used. They simply assume the definition given to them. Understanding is not important. I think especially of idiots like Zelos Malum and "Me".

> > (2) Who has decided that this is not contradiction in set theory?
> It is definitely a contradiction. I have decided it is a contradiction a long time ago. Your comment sums it up quite nicely:
> Meanwhile it has turned out that the set of all real numbers is countable but not listable because then the diagonalization would produce another real number but not listed real number.
> I presume you use the binary or decimal tree to show that these "reals" in the set (0,1) are indeed countable?

That is a way to show the countability of all real numbers. But for the constructible real numbers the not listability has been recognized by someone before us. That would have been the ultimate argument that set theory is nonsense. But set theorists appear to be unable even to remotely consider this possibility. It is easier to convince a stone of logic than a set theorist.

Regards, WM

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