Re: Platonism

From: Mitch Harris (harrisq_at_tcs.inf.tu-dresden.de)
Date: 12/06/04 

Date: 6 Dec 2004 20:31:23 GMT

Lester Zick <lesterDELzick@worldnet.att.net> wrote:
>On Mon, 06 Dec 2004 10:03:10 +0100, Mitch Harris wrote:
>>Lester Zick wrote:
>> >
>>> I don't see how first, second, third, etc. can possibly map one to one
>>> to one, two, three because the intervals involved cannot be assumed
>>> the same, which is the whole point of cardinal numbers.
>>
...
>>The ordinals of two sets are the same if there is a one-to-one
>>order-preserving correspondence between the two sets (and associated
>>-total- orderings; you can't compare two sets ordinally without an
>>order).
>>
>>So technically, cardinals are not ordinals.
>
>OK. Good. Some progress, at least.
>
>>But, consider a finite ordinal, say one with three elements (yes,
>>having cardinality 3). Then, for all six possible total orderings of
>>these three elements (the 6 possible ways of making an ordinal out of
>>it), they is an obvious order preserving isomorphism, between any pair
>>(lowest to lowest, middle to middle, highest to highest).
>>
>>So there is exactly one ordinal for any finite cardinal. Is the finite
>>ordinal identical to its corresponding finite cardinal? Technically,
>>no, but the correspondence tells you they act pretty much the same and
>>can be easily converted one to the other.
>
>But only preserving isomorphism. Technically they are not the same
>unless isomorphism is preserved, and practically if isomorphism is
>preserved you are only discussing cardinal sets to begin with. When
>discussing ordinality in general, there is no necessary isomorphism
>and no necessary cardinality. But I appreciate the straightforward
>reply.

For finite ordinals and cardinals there is an obvious necessary
isomorphism (which involves forgetting the order).

In general, well, that's another story.

Mitch

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: Platonism
    ... >>it), they is an obvious order preserving isomorphism, between any pair ... >But only preserving isomorphism. ... For finite ordinals and cardinals there is an obvious necessary ...
    (sci.math)