Cut‐conditions on sets of multiple‐alternative inferences

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set‐theoretic assumptions, to what I will call the Cut‐for‐Formulas to Cut‐for‐Sets Theorem: for a set F and a binary relation ⊢ on P(F)$\mathcal {P}(F)$, if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order‐theoretic variant of the Tukey‐Teichmüller Lemma. I then discuss relationships between various cut‐conditions in the absence of finitariness or of monotonicity.