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 will call Cut-for-Formulas Cut-for-Sets Theorem: for a set F and binary relation ⊢ on P ( ) $\mathcal {P}(F)$ , if finitary, monotonic, satisfies cut formulas, then it also sets. deduce CF/CS from Ultrafilter twice; each proof uses different order-theoretic variant of Tukey-Teichmüller Lemma. discuss relationships between various cut-conditions in absence finitariness or monotonicity.