Cut-conditions on sets of multiple-alternative inferences

Mathematical Logic Quarterly 68 (1):95 - 106 (2022)
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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 Power(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- Teichm├╝ller Lemma. I then discuss relationships between various cut-conditions in the absence of finitariness or of monotonicity.

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Harold Hodes
Cornell University


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