I want to sketch here the proof that if is a sequence of finite nonempty sets, and then has size for any nonprincipal ultrafilter on
The argument I present is due to Frayne, Morel, Scott, Reduced direct products, Fundamenta Mathematica, 51 (1962), 195–228.
The topic of the size of ultraproducts is very delicate and some open questions remain. For ultraproducts of finite structures, this is continued in Keisler, Ultraproducts of finite sets, The Journal of Symbolic Logic, 32 (1967), 47–57, and finally in Shelah, On the cardinality of ultraproduct of finite sets, The Journal of Symbolic Logic, 35 (1) (Mar., 1970), 83–84. Shelah shows that if an ultraproduct of finite sets is infinite, say of size then His argument is a very nice application of non-standard analysis. The case that interests us is easier.
so it suffices to show that
We need the following combinatorial lemma:
Lemma 1 There is a family of functions such that:
- For any and any and
- If are in then is finite.
Proof: For each let be given by
where is the characteristic function of i.e., if and else
Then is as needed.
Let so the sets are all finite and partition For each let be a list of distinct elements of
Let be as in the lemma. For let be given by
where is such that
Note that if are in then
is a finite union of finite sets and therefore finite. Hence, and we are done.
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