Remarks on Martin's Axiom and the Continuum Hypothesis

Todorcevic, Stevo

Canadian journal of mathematics, Tome 43 (1991) no. 4, pp. 832-851

Résumé

Martin's axiom and the Continuum Hypothesis are studied here using the notion of a ccc partition i.e., a partition of the form where K0 has the following properties:(a) K0 contains subsets of its elements as well as all singletons of X.(b) Every uncountable subset of K0 contains two elements whose union is in K0.

Détail

DOI : 10.4153/CJM-1991-048-8

Mots-clés : 03E05, 03E50, 04A20

Bibliographie

[0] 0. Baumgartner, J.E. and Taylor, A.D., Saturation Properties of Ideals in Generic Extensions H, Trans. Amer. Math. Soc. 271(1982), 587–609. Google Scholar

[1] 1. Comfort, W.W. and Negrepontis, S., Chain Conditions in Topology. Cambridge University Press, 1982. Google Scholar

[2] 2. Devlin, K.J., Some weak versions of large cardinal axioms, Ann. Math. Logic 5(1973), 291–325. Google Scholar

[3] 3. van Douwen, E.K. and Fleissner, W.G., Definable fore ing axiom: an alternative to Martin's axiom, Topology and its Appl. 35(1990), 277–289. Google Scholar

[4] 4. Fremlin, D.H., Consequences of Martin's axiom. Cambridge University press, 1984. Google Scholar

[5] 5. Harrington, L. and Shelah, S., Some exact equiconsistency results in set theory, Notre Dame J. Formal Logic 26(1985), 178–188. Google Scholar

[6] 6. Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory. Higher Set Theory (Miiller, G.H. and Scott, D.S. eds.) Lecture Notes in Math. 669, Springer-Verlag, Berlin-Heidelberg-New York, 1978. Google Scholar

[7] 7. Kanamori, A., Partition relations for successor cardinals, Advances in Math. 59(1986), 152–169. Google Scholar

[8] 8. Knaster, B., Sur une propriété characteristique de l'ensemble des nombres réels, Mat. Sbornik 16(1945), 281–288. Google Scholar

[9] 9. Knaster, B. and Szpilrajn, E., Problem 192. May 1941, The Scottish Book. Google Scholar

[10] 10. Koppelberg, S., General Theory of Boolean Algebras. Vol 1 of Handbook of Boolean algebras (J.D. Monk éd.), North-Holland, Amsterdam, 1989. Google Scholar

[11] 11. Marczewski, E., Séparabilité et multiplication cartésienne des espaces topologiques, Fund. Math. 34(1947), 127–143. Google Scholar

[12] 12. Martin, D.A. and Solovay, R.M., Internai Cohen extensions, Ann. Math. Logic 2(1970), 143–178. Google Scholar

[13] 13. Merrill, J.W.L., UFA fails in the Bell-Kunen model, J. Symbolic Logic 55(1990), 284–296. Google Scholar

[14] 14. Miller, A.W. and Prikry, K., When the continuum has confinality ω , Pacific J. Math. 115(1984), 399–407. Google Scholar

[15] 15. Roitman, J., A very thin superatomic Boolean algebra, Algebra Universalis 21(1985), 137–142. Google Scholar

[16] 16. Shanin, N.A., On intersection of open subsets in the product of topological spaces, Comptes Rendus Acad. Sci. U.R.S.S. 53(1946), 449–501. Google Scholar

[17] 17. Shoenfield, J.R., The problem of pre die ativity. Esays on the Foundations of Mathematics, The Mages Press, Jerusalem, 1961,132–142. Google Scholar

[18] 18. Solovay, R.M., A model of set-theory in which every set of reals is Lebesque measurable, Ann. Math. 92(1970), 1–56. Google Scholar

[19] 19. Todorcevie, S., Remarks on cellularity in products, Compositio Math. 57(1986), 357–372. Google Scholar

[20] 20. Todorcevie, S. Partitioning pairs of countable ordinals, Acta Mathematica 159(1987), 261–294. Google Scholar

[21] 21. Todorcevie, S. Partition Problems in Topology. American Mathematical Society, Providence, 1989. Google Scholar

[22] 22. Todorcevic, S. and Velickovic, B., Martin's axiom and partitions, Compositio Math. 63(1987), 391–408. Google Scholar