Re: Han's startling new set theory.

On Mon, 15 Aug 2005 11:54:37 GMT, Martin Shobe wrote:
> On 14 Aug 2005 16:53:15 -0700, cbrown@xxxxxxxxxxxxxxxxx wrote:
>>(A question I have is: "We have this collection of cardinals, and we
>>assume that the collection is such that every set can be bijected to
>>/at most/ one cardinal in that collection. But how do we know that /at
>>least/ one cardinal exists for every set of interest?")

> I've never really thought about this because I usually consider the
> cardinals to be the equivalence classes. I.e. The fact that the set
> exists is enough. However, I do believe that one definition of the
> cardinal numbers is the smallest ordinal in one of the equivalence
> classes, so proving that one exists for each equivalence class should
> do it.

> One might be able to do something like, well order the set. Map that
> set onto an ordinal, and then prove that any equivalence class that
> has on ordinal, has a smallest one.

If we assume the axiom of choice, then every set can be well ordered.
This is equivalent to saying that every set can be bijectively mapped to
some ordinal. We just pick the smallest such ordinal, and call that the
cardinality of the set.


--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.

Relevant Pages

  • Re: Need Graph Isomorphism Algorithm De-bunked
    ... each component of the graph should be considered as a second ... equivalence class over the vertices and be intersected with the first. ... vector of size N containing equivalence classes for the corresponding ...
    (sci.crypt)
  • Re: Testing for the equivalence relation
    ... >> There are just two equivalence classes. ... assigned to an equivalence class. ... Depending on the nature of the equivalence relation, ...
    (comp.databases.theory)
  • Re: Rational Numbers and Equivalence Classes
    ... each rational number _is_ an equivalence class. ... single equivalence class. ... I doubt, however, that equivalence classes are at all necessary. ...
    (sci.math)
  • Re: The naive test for equality
    ... >> Then rationals are a set Q of equivalence classes defined by the above ... The usual definition of the equivalence class goes is: ... and they may be several of these representations for one value. ...
    (comp.databases.theory)
  • Re: An uncountable countable set
    ... equivalence class and the equivalence class itself. ... This is the principle used in NBG using the von Neumann naturals as ... subjectivity, but the fact that any normal theory only addresses certain ...
    (sci.math)