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Martin's Maximum, Saturated Ideals, and Non-Regular Ultrafilters. Part I

M. Foreman, M. Magidor and S. Shelah
Annals of Mathematics
Second Series, Vol. 127, No. 1 (Jan., 1988), pp. 1-47
Published by: Annals of Mathematics
DOI: 10.2307/1971415
Stable URL: http://www.jstor.org/stable/1971415
Page Count: 47
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Martin's Maximum, Saturated Ideals, and Non-Regular Ultrafilters. Part I
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Abstract

The authors present a provably strongest form of Martin's axiom, called Martin's Maximum, and show its consistency. From it we derive the solutions to several classical problems in set theory, showing that 2ℵ 0 = ℵ2, the non-stationary ideal on ω1 is ℵ2-saturated, and several other results. We show as a consequence of our techniques that there can be no "nice" inner model of a supercompact cardinal. We generalize our results to cardinals above ω1 to show, for example, the consistency of the statement "The non-stationary ideal on every regular cardinal κ is precipitous."

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