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Precipitous towers of normal filters

  • Douglas R. Burke (a1)


In this paper we investigate towers of normal filters. These towers were first used by Woodin (see [15]). Woodin proved that if δ is a Woodin cardinal and P is the full stationary tower up to δ (P) or the countable version (Q), then the generic ultrapower is closed under < δ sequences (so the generic ultrapower is well-founded) ([14]). We show that if ℙ is a tower of height δ, δ supercompact, and the filters generating ℙ are the club filter restricted to a stationary set, then the generic ultrapower is well-founded (ℙ is precipitous). We also give some examples of non-precipitous towers. We also show that every normal filter can be extended to a V-ultrafilter with well-founded ultrapower in some generic extension of V (assuming large cardinals). Similarly for any tower of inaccessible height. This is accomplished by showing that there is a stationary set that projects to the filter or the tower and then forcing with P below this stationary set.

An important idea in our proof of precipitousness (Theorem 6.4) has the following form in Woodin's proof. If are maximal antichains (i Є ω and δ Woodin) then there is a κ < δ such that each AiVκ is semiproper, i.e.,

contains a club (relative to ∣ a∣ < κ).



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[1]Burke, D., Splitting stationary subsets of , preprint.
[2]Foreman, M., Potent axioms, Transactions of the American Mathematical Society, vol. 294 (1986), pp. 128.
[3]Foreman, M., Magidor, M., and Shelah, S., Martins maximum, saturated ideals and non-regular ultrafilters. Part I, Annals of Mathematics, vol. 127 (1988), pp. 147.
[4]Jech, T., Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5 (1973), pp. 165198.
[5]Jech, T., Set theory, Academic Press, 1978.
[6]Jech, T., Some properties of δ-complete ideals defined in terms of infinite games, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 3145.
[7]Jech, T., Magidor, M., Mitchell, W., and Prikrv, K., Precipitous ideals, this Journal, vol. 45 (1980), pp. 18.
[8]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Lecture notes in mathematics, vol. 669, 1978, pp. 99275.
[9]Kunen, Kenneth, Set theory, an introduction to independence proofs, North-Holland, 1980.
[10]Martin, D. A., Borel and projective games, forthcoming book.
[11]Martin, D. A. and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 4 (1989), pp. 71125.
[12]Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Lecture notes in logic, vol. 3, Springer-Verlag, 1991.
[13]Woodin, H., absoluteness and supercompact cardinals, some notes, 05 15, 1985.
[14]Woodin, H., Lectures at UCLA-Caltech logic seminar, 19851986.
[15]Woodin, H., Supercompact cardinals, sets of reals and weakly homogeneous trees, Proceedings of the National Academy of Sciences of the United States of America, vol. 85 (1988), pp. 65876591.

Precipitous towers of normal filters

  • Douglas R. Burke (a1)


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