> On 7 Feb., 15:49, fom <fomJ...@nyms.net> wrote: >> On 2/7/2013 7:54 AM, WM wrote: >> >> > On 7 Feb., 09:10, William Hughes <wpihug...@gmail.com> wrote: >> >> On Feb 7, 9:00 am, WM <mueck...@rz.fh-augsburg.de> wrote: >> >> <snip> >> >> >>> What does that mean for the set of accessible numbers? >> >> >> That this potentially infinite set is not listable. >> >> > Here we stand firm on the grounds of set theory. >> >> > Once upon a time there used to be a logocal identity: The expression >> > "Set X is countable" used to be equivalent to "Set X can be listed". >> >> Incorrect. > > First learn, then understand, then judge. > Otherwise you appear silly. > > 1.1 Definition The cardinality of A is less than or equal to the > cardinality of B (notation: |A| ? |B|) if there is a one-to-one > mapping of A into B. > > Notice that |A| ? |B| means that |A| = |C| for some subset C of B. We > also write |A| < |B| to mean that |A| ? |B| and not |A| = |B|, i.e., > that there is a one-to-one mapping of A onto a subset of B, but there > is no one-to-one mapping of A onto B. Notice that this is not at all > the same thing as saying that there exists a one-to-one mapping of A > onto a proper subset of B: for example, there exists a one-to-one > mapping of the set N onto its proper subset (Exercise 2.3 in Chapter > 3), while of course |N| = |N|. > It was pointed out in Chapter 3 that the property |A| = |B| behaves > like an equivalence relation: it is reflexive, symmetric, and > transitive. We show next that the property |A| ? |B| behaves like an > ordering on the "equivalence classes" under equipotence. > 1.2 Lemma > (a) If |A| ? |B| and |A| = |C|, then |C| ? |B|. > (b) If |A| ? |B| and |B| = |C|, then |A| ? |C|. > (c) |A| ? |A|. > (d) If |A| ? |B| and |B| ? |C|, then |A| ? |C|. > > Proof. Immediate from definition.? > > [Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" Marcel > Dekker Inc., New York, 1984, 2nd edition.] > > Regards, WM
Your assumption that fom is in need to learn such basics is another indication of your complete ignorane and stupidity.