Core model: Difference between revisions

Revision as of 08:12, 27 October 2014

Revision as of 18:42, 3 February 2015

Line 28:

Partial results for the conjecture are that:

#If there is no inner model with a Woodin cardinal, then K^c exists and is fully iterable.

#If (boldface) Σ¹_n+1 determinacy (n is finite) holds in every generic extension of V, but there is no iterable inner model with n Woodin cardinals, then K exists.

#If there is a measurable cardinal κ, then either K^c below κ exists, or there is an ω₁+1 iterable model with measurable limit λ of both Woodin cardinals and cardinals strong up to λ.

#If there is noa innermeasurable modelcardinal withκ, athen strongeither cardinalK^c thatbelow κ exists, or there is aan ω₁+1 iterable model with measurable limit λ of strongboth Woodin cardinals, thenand Kcardinals existsstrong up to λ.

#If every set has a sharp or there is a proper class of subtle cardinals, but there is no inner model with a Woodin cardinal, then K exists.

#If (boldface) Σ¹_n+1 determinacy (n is finite) holds in every generic extension of V, but there is no iterable inner model with n Woodin cardinals, then K exists.

If V has Woodin cardinals but not cardinals strong past a Woodin one, then under appropriate circumstances (a candidate for) K can be constructed by constructing K below each Woodin cardinal (and below the class of all ordinals) κ but above that K as constructed below the supremum of Woodin cardinals below κ. The candidate core model is not fully iterable (iterability fails at Woodin cardinals) or generically absolute, but otherwise behaves like K.

Line 39 ⟶ 37:

* W.H. Woodin (2001). [http://www.aimath.org/WWN/coremodel/coremodel.pdf The Continuum Hypothesis, Part I]. Notices of the AMS.

* William Mitchell. "Beginning Inner Model Theory" (being Chapter 17 in Volume 3 of "Handbook of Set Theory") at [http://www.math.ufl.edu/~wjm/papers/]{{dead link|date=April 2014}}.

* Matthew Foreman and Akihiro Kanamori (Editors). "Handbook of Set Theory", Springer Verlag, 2010, ISBN 978-1402048432.

* Ronald Jensen and John Steel. "K without the measurable". Journal of Symbolic Logic Volume 78, Issue 3 (2013), 708-734.

[[Category:Inner model theory]]