Re: Platonism

From: Nathan (ntspam2_at_netscape.net)
Date: 12/09/04 

Date: 9 Dec 2004 10:14:22 -0800

tchow@lsa.umich.edu wrote:
> In article <41b49da7$21$fuzhry+tra$mr2ice@news.patriot.net>,
> Shmuel (Seymour J.) Metz <spamtrap@library.lspace.org.invalid> wrote:
> <In <41b39de4$0$565$b45e6eb0@senator-bedfellow.mit.edu>, on
12/05/2004
> < at 11:46 PM, tchow@lsa.umich.edu said:
> <
> <>For example, in some context where you're studying ordered sets, it
> <>may be good to define an ordered set as a set equipped with a
> <>relation "<" satisfying certain conditions. The ordinal 2 would be
a
> <>special case of this, and so wouldn't be the von Neumann ordinal
but
> <>a set with 2 elements equipped with a total ordering.
> <
> <No. The ordered sets (1,2) and (2,1) have the same order type, that
> <given by the ordinal 2. In fact, any two finite ordered sets with
the
> <same cardinality will have the same order type.
>
> What I meant was, you might choose some set with 2 elements, equipped
> with a total ordering, as your canonical representative of the class
of
> well-ordered sets with 2 elements. There is nothing that "forces"
you
> to always choose the von Neumann ordinal to be your canonical
> representative.

I still don't see your point. You said that the von Neumann definition
of ordinals fails to capture some aspects of the ordinal concept. A von
Neumann ordinal is a well-ordered set with order defined by set
membership. Granted, many other sets with equivalent order relations
can be defined, but the differences between these well-ordered sets are
irrelevant to the concept of an ordinal. They're all models of the same
axioms, as it were.

Which part of the ordinal concept do they lack?

Relevant Pages

  • Re: proof of uncountable von Neumann ordinals in ZFC
    ... The von Neumann definition of an ordinal ... Axiom Schema of Replacement states: ... go from a well-ordered set to a von Neumann ... Given a well-ordered set x, let a be the supremum of all ordinals that are order-isomorphic to either x or an initial segment of x. ...
    (sci.math)
  • Re: is there a set of all ordinals
    ... of all Von Neumann ordinals; ... we don't have a proper class of all VON NEUMANN ... Every ordinal has a successor. ...
    (sci.logic)
  • Re: is there a set of all ordinals
    ... of all Von Neumann ordinals; ... What do you mean there is a proper class of all Von Neumann ordinals, ... not a Von Neumann ordinal then it is not be a member in it. ...
    (sci.logic)
  • Re: proof of uncountable von Neumann ordinals in ZFC
    ... The von Neumann definition of an ordinal ... Axiom Schema of Replacement states: ... go from a well-ordered set to a von Neumann ... Given a well-ordered set x, let a be the supremum of all ordinals that are order-isomorphic to either x or an initial segment of x. ...
    (sci.math)
  • Re: "Proper functions", Replacement and Zorns Lemma.
    ... no (this is because Zermelo set theory holds in ... V_omega+omega where all von Neumann ordinals are below omega+omega). ... What difficulty do you see in giving a precise statement of replacement? ...
    (sci.logic)
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