502 – Ultraproducts of finite sets

October 2, 2009

I want to sketch here the proof that if {(M_n\mid n\in{\mathbb N})} is a sequence of finite nonempty sets, and {\lim_n |M_n|=\infty,} then {\prod_nM_n/{\mathcal U}} has size {|{\mathbb R}|} for any nonprincipal ultrafilter {{\mathcal U}} on {{\mathbb N}.}

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 {\kappa,} then {\kappa^{\aleph_0}=\kappa.} His argument is a very nice application of non-standard analysis. The case that interests us is easier.

Clearly,

\displaystyle |\prod_nM_n/{\mathcal U}|\le|\prod_n M_n|\le|{\mathbb N}^{\mathbb N}|=|{\mathbb R}|,

so it suffices to show that {|{\mathbb R}|\le|\prod_nM_n/{\mathcal U}|.}

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