Abstract.
We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)+ and (rTT)++; each is equivalent to (rTT) in ZF. The variation (rTT)++ applies rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado.
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Dedicated to W.W. Comfort on the occasion of his seventieth birthday.
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Hodel, R. Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem. Arch. Math. Logic 44, 459–472 (2005). https://doi.org/10.1007/s00153-004-0264-9
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DOI: https://doi.org/10.1007/s00153-004-0264-9