 For infinite M the collection of subsets of \(M^n\) and the set of functions \(M^n\to M\) are famously complex. Currently set theory (ZFC) cannot even decide how many subsets of an infinite set there are. The situation is quite different with first order logic. The respective concept “for some \(a \in M\)” is unproblematic, relatively speaking. Of course, depending on what M is, “for some \(a \in M\)” can be quite problematic too. To reflect the difficulties involved with finding a \(P \subseteq M^n\) or an \(f:M^n \to M\) we sometimes say we “guess” a \(P \subseteq M^n\) or an \(f:M^n \to M\).