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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 UTC7, WM wrote: > > > Am Montag, 2. Oktober 2017 00:39:51 UTC+2 schrieb John Dawkins: > > > > In article <fb18e185529b428892d818268a0912cc@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".



