With Ramsey-theoretic methods we show: It is consistent that there is a forcing that diagonalises one ultrafilter over ω and preserves another ultrafilter.

Logics for ‘generally’ were introduced for handling assertions with vague notions, such as ‘generally’, ‘most’, ‘several’, etc., by generalized quantifiers, ultrafilter logic being an interesting case. Here, we show that ultrafilter logic can be faithfully embedded into a first-order theory of certain functions, called coherent. We also use generic functions (akin to Skolem functions) to enable...

Proof: By assumption, φinf is preserved under ultrafilter extensions. The second conjunct of θ, i.e., ∀xy.(x = y → Rxy) is also preserved under ultrafilter extension, since it is modally definable using global modality. Finally, consider the third conjunct of θ. From the fact that M |= ∀xy.(x = y → Rxy), we can derive that the denotation of R in ueM includes all pairs of ultrafilters (u, v) suc...

Via two short proofs and three constructions, we show how to increase the model-theoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regul...

A set X ⊆ 2 has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s)...

A set X ⊆ 2 has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s)...