Re: Platonism

From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 12/03/04

  • Next message: Dave Seaman: "Re: Platonism" 
    Date: Fri, 03 Dec 2004 20:03:59 GMT

    On 03 Dec 2004 18:46:06 GMT, tchow@lsa.umich.edu in comp.ai.philosophy
    wrote:

    >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.

    Well, I have to apologize here as I have to Dave for what is obviously
    a confusion of terms. I thought the idea of ordinality was pretty
    clear, but based on what I read in Dave's references, there is plainly
    a different definition for ordinal numbers in math in which finite
    ordinality and cardinality are the same.

    >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?

    Well, I appreciate what you're asking. My question concerned whether
    natural numbers include both cardinal numbers (which I take to be
    those like one, two, three, etc. for which differences between
    successive cardinal numbers are identical) and ordinal numbers (which
    I take to represent those like first, second, third, etc. for which
    intervals are not necessarily the same)?

    I can see now that there is a difference in terminology that made the
    original question ambiguous. My remark concerning a contemporary math
    tutorial was just intended to avoid unnecessary confusion. But it is
    apparent that finite ordinality and cardinality would be the same
    where these terms are used to describe a cardinal set to begin with. I
    suppose my question really reduces to whether ordinals not used to
    describe a cardinal set are natural numbers?

    Regards - Lester

  • Next message: Dave Seaman: "Re: Platonism"

    Relevant Pages

    • Pure-cardinal approach *is* possible! (was: Mathematical concepts)
      ... Your ordinals start with zeroth, not first, right? ... So cardinals are defined, in terms of sets of ordinals, like ... For example if you think of your head, your hands, suits ... there's a gap between hands and suits. ...
      (sci.math)
    • Re: Pure-cardinal approach *is* possible! (was: Mathematical concepts)
      ... >> it is necessary to distinguish finite from infinite. ... >ordinals start with 1, just as the ancients did, and only introduce ... >zero after the negatives have already been introduced. ... So cardinals are defined, in terms of sets of ordinals, like ...
      (sci.math)
    • Re: Mathematical concepts
      ... >concept of counting to pre-schoolers. ... How does one KNOW that finite cardinals make sense? ... For ordinals, even infinite ordinals, it is ... >> Counting on fingers is ordinal. ...
      (sci.math)
    • Re: Mathematical concepts
      ... >> Counting is ordinal, not cardinal. ... correspondence between the two for finite ordinals and ... cardinals, preserving the arithmetic operations, is ... Teachers need to be educated, ...
      (sci.math)
    • Re: Amateur continuum hypothesis question
      ... There are two common kinds of transfinite "number": ordinals and cardinals. ... The well-ordering theorem I stated above implies ... correspondence with an initial segement of Y. The basic property I ...
      (sci.math)