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Re: Intuitive criterion for set size.
Posted:
Jan 12, 2013 4:32 PM
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On Jan 12, 2:08 am, Zuhair <zaljo...@gmail.com> wrote:
> For any two sets A,B:
>
> [1] A is bigger than B iff |A| > |B| Or A,B are sets of naturals and
> there exist a set C of naturals such that for every element n of C: |
> A(n)| > |B(n)| and |A(n+1)| - |B(n+1)| > |A(n)| -|B(n)|.
>
> where X(n) = {y| y in X & y <' n};
> <' stands for natural strict smaller than relation;
> | | stands for cardinality defined after Cantor's.
According to this definition, if A and B are *any* sets of naturals,
then A is bigger than B (and B is bigger than A). Perhaps, by "a set C
of naturals", you meant "a nonempty set C of naturals"?
By the way, your condition
"|A(n+1)| - |B(n+1)| > |A(n)| -|B(n)|"
is equivalent to the simpler statement
"n is an element of A but not an element of B".
By saying that this condition holds for every element n of C, you are
in effect saying that C is a subset of A and is disjoint from B, i.e.,
C is a subset of the set difference A\B.
> [2] A is smaller than B iff B is bigger than A.
>
> [3] A is equinumerous to B iff ~ A bigger than B & ~ A smaller than B.
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