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From: Steffen Schuler on 27 Jan 2010 23:36
Hi,

I found and proved the following cardinal number theorem inspired by
exercise 35 on page 165 in
H. B. Enderton: Elements of Set Theory, Academic Press, 1977.

Let Nat be the set of natural numbers including 0.
Let Aleph0 be the cardinality of Nat.
For a set M let Pot(M) be the set of subsets of M
and let card(M) be the cardinality of M.
Let A@B be the intersection of the sets A and B.
Let b^a be the cardinality of the set of functions from A to B
where a is the cardinality of A and b the cardinality of B.


Theorem:
For a set L of natural numbers let M(L) be the set of subsets A
of Pot(Nat) with

For all distinct B and C in A: card(B@C) is in L.

and let K(L) be the set of card(A) for A in M(L).

If L is a finite nonempty subset of Nat, then Aleph0 is the maximum of
K(L). And if L is a infinite subset of Nat, then 2^Aleph0 is the maximum
of K(L).


(There are easy proofs which need only the axioms from ZFC.)

Have fun in proving that!

Steffen Schuler


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