"WM" <mueckenh@rz.fh-augsburg.de> wrote in message news:ccf88754-1d64-46f3-ad9f-372ef6fe91c9@i28g2000yqa.googlegroups.com... > On 22 Jun., 23:21, "Mike Terry" > <news.dead.person.sto...@darjeeling.plus.com> wrote: > > > > > To spell it out clearly: The set of all diagonals (including or > > > exluding all rationals - that does not matter) > > > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > > > that are constrcuted according to my prescription cannot be listed > > > although it is countable. > > > > Yes, that's clear, thanks! > > Youe welcome. > > > > But of course the set can be listed: > > > > Â Â (A0, L0(0), A1, L0(1), A2, L0(2), A3, L0(3), ...) > > > > and the set is countable. > > a) Does this list contain the anti-diagonal of > (..., An, ... A2, A1, A0, L0)?
This is not a list of numbers. L0 is not a number, it is a list.
Therefore (..., An, ... A2, A1, A0, L0) does not have an anti-diagonal.
> Then Cantor's argument fails as a list contains its antidiagonal. > Reason: you list in your listing above only the lines of lists. That > is so because the antidiagonal of every list Ln belongs to another > list L(n+1)). > > b) Does it not? Then a list can not list all anti-diagonals that > belong to the countabe set constructed in my argument. > > > > > > > > > > If we use Cantor's definiton of "countable", then the set > > > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > > > is uncountable. > > > > Cantor's definition of countable is that there is an injection from the set > > to N (set of natural numbers). > > > > E.g.: > >   A0   --->   1 > >   L0(0)  --->   2 > >   A1   --->   3 > >   L0(1)  --->   4 > >   A2   --->   5 > >   L0(2)  --->   6 > >   ... > > > > Right? > > No. See above. > Another example ist the set of all definable reals. There is no > bijection with |N. But they belong to a countable set of all finite > words. > > > > > > If we use the definition that a subset of a countable set is > > > countable, then the set > > > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > > > is countable. > > > > That's not a definition, it's a theorem. > > That may be a theorem in ZFC. In mathematics a subset cannot have a > larger cardinality than its superset. > > > > > > > > > > This is wrong.  An obvious listing is (A0, A1, ...) > > > > > The set > > > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > > > cannot be listed. > > > > Obviously it can.  (I'm completely missing why you could possibly be > > thinking it couldn't) > > You will understand if you try to answer question a) above. > > > > > > > > > >   there exists a countable set M, such that > > > >     If L is a Cantor-list, then > > > >       (anti-?)diagonal of L belongs to M. > > > > > > That is so obviously false that its banal. > > > > > No it is not. If there exists a Cantor-list, i.e., that what Cantor > > > really understood by the term list, then it is a list of *defined* > > > reals. And then its anti-diagonal is a defined real too. Then exists a > > > countable set M, namely the set of all defined reals, that is > > > countable. Nevertheless it cannot be listed. > > > > I don't believe that was Cantor's definition of list.  (But I'm prepared to > > be persuaded if you've got some references for this?) > > If you understand enouf´gh German, then you might look here: > > 1906, 8. Aug. letter Cantor to Hilbert > Lieber Freund > ... > König will zwei Arten von reellen Zahlen unterscheiden; solche, die > âendliche Definitionen" zulassen und solche, die âunendliche > Definitionen" erfordern. > Eine jede Definition ist aber ihrem Wesen nach eine endliche, d. h. > sie erklärt den zu bestimmenden Begriff durch eine endliche Anzahl > bereits bekannter Begriffe > B1, B2, B3, ...,Bn. > âUnendliche Definitionen" (die nicht in endlicher Zeit verlaufen) sind > Undinge. > Wäre Königs Satz, daà alle âendlich definirbaren" reellen Zahlen einen > Inbegriff von der Mächtigkeit ï0 ausmachen, richtig, so hieÃe dies, > das ganze Zahlencontinuum sei abzählbar, was doch sicherlich falsch > ist. > ...
I will have a go at that, but I only did "O-level" German 30 or so years ago, so I'm not sure how I'll get on... (Thanks anyway)
> > Essential: Infinite definitions, are nonsense. If there were only > alef_0 definable reals, then the continuum was countable. > > So Cantor clearly states, that undefinable reals are nonsense. And of > course he is right. A number must be in trichotmy with others. > Otherwise it is not a number. The idea of undefinable numbers was > created in order to save set theory, just as some fools now claim > uncountably many languages in order to invalidate my list of all words > in all languages. > > Regards, WM >
