Re: Platonism

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

Date: 03 Dec 2004 21:07:35 GMT

In article <41b16963.2606882@netnews.att.net>,
Lester Zick <lesterDELzick@worldnet.att.net> wrote:
>On Fri, 03 Dec 2004 18:57:18 GMT, patty <pattyNO@SPAMicyberspace.net>
>in comp.ai.philosophy wrote:
>>Point of information: regardless of whether we are talking within the
>>context of contemporary mathematics or within the context of natural
>>language, it seems to me that if we can distinguish between the
>>cardinals and the ordinals, well then sets of those things cannot be
>>identical. But certainly they can map one to one. What am i missing?

Contemporary mathematicians have a habit, perhaps a bad habit, of taking
two things that are the same "for all practical purposes"---where what
counts as "practical" is *context-sensitive*---and declaring them to be
identical. Thus, the ordered pair (x,y) is *identified* with {{x}, {x,y}}
and a function is identified with a set of ordered pairs and two sets with
the same members are said to be *equal*.

There aren't many, if any, contexts in mathematics in which one needs to
make a clear distinction between a finite ordinal and a finite cardinal.
So, they are typically declared to be equal. The set of all finite
ordinals/cardinals is { {}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}, ... }

I can imagine a context in which you want to think about cardinals as
equivalence classes of sets, and ordinals as equivalence classes of
*well-ordered* sets, and then they would be different. But one would
then need to specify the context, and probably introduce custom-made
definitions of "ordinal" and "cardinal," answering the question by fiat.

>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.

I'm lost here, I'm afraid. What intervals are you talking about?
Are you visualizing some kind of interval between one and two? Or
between first and second? I don't understand the character of this
interval, unless you are embedding the numbers into a geometric space. 

-- 
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

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