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UFA fails in the Bell-Kunen model

  • John W. L. Merrill (a1)


In [vDF], van Douwen and Fleissner introduce a number of axioms which hold in models constructed by iteratively forcing MA in a nontrivial extension of the set-theoretic universe. One such model is the Bell-Kunen model, obtained by starting with a model of ZFC + GCH, then forcing “MA + ϲ = ω2” by the standard means, then forcing “MA + ϲ = ω3”, and so on. The Bell-Kunen model is the result of an ω1 sequence of extensions of this form, with direct limits taken at limit ordinals. (See [BK] for a more complete description.) Van Douwen and Fleissner observed that many of the properties of this model could be distilled into a “Definable Forcing Axiom”, which states that “If P is a c.c.c. partial order which is definable from a real, then there is a sequence of filters through , such that if is any dense subset of P, then all but countably many of the ℱα's meet in a nonempty set.” (They call such a sequence ω1-generic.) Van Douwen and Fleissner ask whether one can eliminate the restriction on the c.c.c. order entirely; the resulting axiom (“If P is any c.c.c. partial order of cardinality at most ϲ, then there is a sequence of filters…”) is called the Undefinable Forcing Axiom (UFA).



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[BK]Bell, M. and Kunen, K., On the PI character of ultrafilters, La Société Royale du Canada. L'Académie des Sciences. Comptes Rendues Mathématiques (Mathematical Reports), vol. 3 (1981), pp. 351356.
[vDF]van Douwen, E. and Fleissner, W., The definable forcing axiom (preprint).
[H]Hajnal, A., Proof of a conjecture of S. Ruziewicz, Fundamenta Mathematicae, vol. 50 (1961/1962), pp. 123128.
[J]Jech, T., Set theory, Academic Press, New York, 1978.
[K]Kunen, K., Set theory: an introduction to independence proofs, North-Holland, Amsterdam, 1980.
[M]Merrill, J., A class of consistent anti-Martin's axioms, Pacific Journal of Mathematics (to appear).
[R]Roitman, J., A very thin, thick, superatomic Boolean algebra, Algebra Universalis, vol. 21 (1985), pp. 137142.


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