Re: Platonism

tchow_at_lsa.umich.edu
Date: 12/03/04 

Date: 03 Dec 2004 18:46:06 GMT

In article <41b134cb.73552732@netnews.att.net>,
Lester Zick <lesterDELzick@worldnet.att.net> wrote:
>The last time I looked, ordinals were defined as first, second, third,
>etc. so I'm not sure I follow what you're saying here. I don't see
>that first, second, third, etc. are identical with one, two, three,
>etc. unless the differences between first, second, third, etc. are the
>same as between one, two, three.
>
>(I'm not looking for you to explain contemporary mathematics to me. I
>thought the question regarding ordinals and cardinals was pretty plain
>to begin with.)

I assume that "the question regarding ordinals and cardinals" was your
question as to whether "the set of all natural numbers includes ordinal
numbers or not." The answer in the context of contemporary mathematics
is that it does, for the somewhat trivial reason that finite ordinals are
identified with finite cardinals in contemporary mathematics. But now
you say you don't want contemporary mathematics to be explained to you,
so I'm at a loss as to what your question is.

Are you asking how contemporary mathematicians can possibly equate
finite ordinals and finite cardinals when ordinality and cardinality
are such manifestly different concepts?

Or are you asking whether the sentence "The set of natural numbers
contains ordinals" is true in some context *other than* the context
of contemporary mathematics? If so, could you explain more fully
what this context is, since unfortunately it isn't so plain to those
of us who are used to dealing with contemporary mathematics? 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

Relevant Pages

  • Re: abundance of irrationals!)
    ... As long as there are only finite ordinals n there are ... >> also only finite cardinals Card. ... A finite ordinal marks a position, a finite cardinal marks an amount. ...
    (sci.math)
  • Re: Platonism
    ... >>The last time I looked, ordinals were defined as first, second, third, ... >>(I'm not looking for you to explain contemporary mathematics to me. ... for the somewhat trivial reason that finite ordinals are ... > identified with finite cardinals in contemporary mathematics. ...
    (comp.theory)
  • Re: Platonism
    ... >>The last time I looked, ordinals were defined as first, second, third, ... >>(I'm not looking for you to explain contemporary mathematics to me. ... for the somewhat trivial reason that finite ordinals are ... > identified with finite cardinals in contemporary mathematics. ...
    (sci.math)
  • Re: An uncountable countable set
    ... The set of finite ordinals IS the set of natural numbers. ... All cardinals are ordinals. ... naturals ordinals then that's a pretty good reason to ignore textbook ...
    (sci.math)
  • Re: Why do so many people believe the pessimistic, nihilistic, and
    ... piles of rocks. ... Cardinals are size - how big is the number, and what other numbers might be ... Ordinals are counting - how do you get to the number, ... all the finite ordinals are also cardinals. ...
    (rec.arts.sf.written)