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Re: Matheology § 210
Posted:
Feb 7, 2013 11:42 AM
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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
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