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.]