In article <ccf88754-1d64-46f3-ad9f-372ef6fe91c9@i28g2000yqa.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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)?
If L0 is a list then the question is irrelevant. If you mean {..., An, ...A0, L0(0), L0(1), ...L0(n),...}, then there are all sorts of "anti-diagonals" possible depending on how you reorder that set into a list.
> Then Cantor's argument fails as a list contains its antidiagonal. But you list is not a list of reals, since one of its members is a list of reals rather than a real, and it is only for list of reals that antidiagonals are to be constructed.
> > b) Does it not? Then a list can not list all anti-diagonals that > belong to the countabe set constructed in my argument.
That depends on the rule or rules by which those antidiagonals are to be constructed.
> > > > > > > > > > If we use Cantor's definiton of "countable", then the set > > > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > > > is uncountable.
I can count it by arranging it as the list {L0(0), A0, L0(1),A1, LO(2),A2, ...} > > > > Cantor's definition of countable is that there is an injection from the set > > to N (set of natural numbers). > > > > E.g.: > > Â Â A0 Â Â ---> Â Â 1 > > Â Â L0(0) Â ---> Â Â 2 > > Â Â A1 Â Â ---> Â Â 3 > > Â Â L0(1) Â ---> Â Â 4 > > Â Â A2 Â Â ---> Â Â 5 > > Â Â L0(2) Â ---> Â Â 6 > > Â Â ... > > > > Right? > > No. See above.
If Cantor's definition is not equivalent to the one you just rejected, what is Cantor's definition?
WM continues to moan about what is and is not countable but refuses to give his definition of countable.
If WM's definition of countable for an infinite set is not euivalnt to there existing a bijection between it and the naturals, then WM's is wrong.
> 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.
Irrelevant, as it cannot hold withing any standard set theory..
Every countable set, and every subset of a countable set injects to the set of naturals and the set of naturals surjects onto it. > > > > > > 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.
Which is entirely consistent with having every subset of a countable set countable. And that is a theorem in FOL+ZFC and, as far as I know, all other standard set theories having countable sets. > > > > > > > > > > This is wrong. Â An obvious listing is (A0, A1, ...) > > > > > The set > > > {..., An, ...A0, L0(0), L0(1), ...L0(n),...} > > > cannot be listed. > > > > Obviously it can. Â (I'm completely missing why you could possibly be > > thinking it couldn't) > > You will understand if you try to answer question a) above.
Since one of the elements of your list (..., An, ... A2, A1, A0, L0), is not a real at all but a list of reals, and your listing is not in standard order, the issue of an antidiagonal is irrelevant.
However, it is trivial, as shown above, to list the reals in L0 along with all the anti-diagonals listed and find an antidiagonal to that.
