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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexander, J.: Ordered sets, complexes, and the problem of bicompactifications. Proc. Nat. Acad. Sci. U.S.A. 22, 296–298 (1939)
Banaschewski, B.: The power of the ultrafilter theorem. J. London Math. Soc. 27, 193–202 (1983)
Blass, A.: Prime ideals yield almost maximal ideals. Fund. Math. 127, 57–66 (1986)
Cowen, R.: Some combinatorial theorems equivalent to the prime ideal theorem. Proc. Am. Math. Soc. 41, 268–273 (1973)
Cowen, R.: A Short Proof of Rado’s Lemma. J. Combinatorial Theory (B) 12, 299–300 (1972)
Engeler, E.: Eine Konstruktion von Modellerweiterungen. Z. Math. Logik Grundlagen Math. 5, 126–131 (1959)
Engelking, R.: General Topology. Heldermann Verlag, Berlin, 1989
Feferman, S.: Some applications of the notion of forcing and generic sets. Fund. Math. 56, 325–345 (1965)
Frink, O., Jr.: Representations of Boolean algebras. Bull. Amer. Math. Soc. 47, 755–756 (1941)
Halpern, J.D., Lévy, A.: The Boolean prime ideal theorem does not imply the axiom of choice. Axiomatic Set Theory, Proc. Sym. Pure Math. 13(part 1), 83–134; AMS, Providence, R.I. 1971
Henkin, L.: Metamathematical theorems equivalent to the prime ideal theorem for Boolean algebras. Bull. Amer. Math. Soc. 60, 387–388 (1954)
Howard, P., Rubin, J.: Consequences of the Axiom of Choice. Mathematical Surveys and Monographs 59, Am. Math. Soc. 1998
Jech, T.: The Axiom of Choice. North Holland, Amsterdam, 1973
Kelley, J.: General Topology. D. Van Norstrand, Princeton, NJ, 1955
Koppelberg, S.: Handbook of Boolean Algebras, vol. 1. D. Monk and R. Bonnet, (ed.), North Holland, Amsterdam, 1989
Parovicenko, I.: Topological equivalents of the Tihonov theorem. Soviet Math. Dokl. 10, 33–34 (1969)
Rado, R.: Axiomatic treatment of rank in infinite sets. Canad. J. Math. 1, 337–343 (1949)
Rav, Y.: Variants of Rado’s Selection Lemma and their Appl. Math. Nachr. 79, 145–165 (1977)
Rubin, H., Rubin, J.: Equivalents of the Axiom of Choice, II. North-Holland, Amsterdam, 1985
Schechter, E.: Handbook of Analysis and Its Foundations. Academic Press, New York, 1997
Shoenfield, J.: Mathematical Logic. Addison-Wesley, Reading, Mass. 1967
Stone, M.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936)
Teichmüller, O.: Braucht der Algebraiker das Auswahlaxiom?. Deutsche Math. 4, 567–577 (1939)
Tukey, J.: Convergence and uniformity in topology. Ann. Math. Studies 2, Princeton University Press, 1940
Wolk, S.: On Theorems of Tychonoff, Alexander, and R. Rado. Proc. Am. Math. Soc. 18, 113–115 (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to W.W. Comfort on the occasion of his seventieth birthday.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-004-0264-9