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Re: Matheology § 210
Posted:
Feb 8, 2013 12:27 PM
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WM wrote:
> 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.
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