From: Lee Corbin (lcorbin@tsoft.com)
Date: Sun Jun 16 2002 - 07:36:43 MDT
Scerir writes
> In example (adapted from a recent discussion on FOM) ...
>
> The theorem that the set of all natural numbers {0,1,...} is
> equinumerous with the set of all even numbers (0,2,...}, is
> true if it is meant that every number has a double, and every
> double is the double of one, and no more than one number.
>
> But if every number has a double, and every double is the
> double of just one number, does it follow that there are "as many"
> doubles (even numbers) as singles (integers), given the meaning
> of "as many"? Is it correct to define "as many" or
> "equinumerous" using the idea of 1-1 correspondence?
The way that I understand it, Cantor really got somewhere with
his definitions. Some people even called it a paradise ;-)
Isn't that how it always is in mathematics? We adopt as fruitful
those axioms that produce interesting results. Cantor's very
sensible definitions allowed one to address bigger collections
(MUCH bigger). My own feeling is that, everything else equal,
bigger is better.
> The argument usually then turns to the challenge of defining
> "as many as". This brings us to argue that we can think of number
> as satisfying n-place predicates such as "x is a different thing
> from y", which is - in general - not satisfied.
Could you elaborate? Now for all numbers x and y, that predicate
is indeed satisfied, isn't it? (Of course, some number, e.g., the
rational number 1/2 can have other *names* such as 2/4 or 3/6, but
we're talking entities here, not names.)
> Any two collections are equinumerous (have as many objects as
> each other) when there is such a predicate they both satisfy.
> Defined in this way, no proper subset can be equinumerous
> with its parent.
I don't understand.
> The parent, by definition, contains objects
> "different" from any of those in the subset, hence the parent
> cannot satisfy the same n-place predicate as the subset.
Yes, clearly "the integers" contain items not in "the doubles",
but what n-place predicate does the parent collection (the
integers) fail to satisfy that the subset collection does
satisfy?
> Another deep example is at
> http://community.middlebury.edu/~arthur/LeibCant.pdf
Yes, I'll read that.
Lee
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