In [KwoM] it is proved that the the core model K exists under the assumption that there is no inner model having a Woodin cardinal and satisfying ZFC. Furthermore, they also mention that the result is the "best possible", in the sense that we cannot weaken this anti-large-cardinal hypothesis without strengthening the remainder of the hypothesis.


From what I understand, using the core model induction (which I must admit I don't know the details of yet), K can be constructed below any finite number of Woodin cardinals. So my question is then whether or not K has been constructed below $\omega$ many Woodins and if so, what the extra hypothesis is? If this is indeed the case, I would greatly appreciate a reference to the given paper.


*[KwoM] Jensen and Steel: "K without the measurable"*


**Edit:** In the introduction to [Schindler and Steel's "The Core Model Induction"](http://wwwmath.uni-muenster.de/logik/Personen/rds/core_model_induction.pdf), they write that if we assume that there is no proper class model with $n+1$ Woodins and over every set there is a proper class model with $n$ Woodins, then $K$ exists (in the sense of the above [KwoM]).


Can this be extended to: if there is no proper class model with $\omega$ Woodins and over every set there is a proper class model with $n$ Woodins, for every $n<\omega$, then $K$ exists?



Core model with $\omega$ Woodin cardinals
Dan, the answer is "no." You might be in a situation where Kc is a mouse (i.e., its ctble. substructures are ω1+1 iterable), but Kc is not fully iterable. E.g. it could be that Kc has M2F,#, where F is the operator simultaneously closing under all Mn#, n<ω, but for some θ, V = LF(Vθ) (same F, V could even be an extender model): the Q-structures needed to fully iterate M2F,# would be given by the M1F,# operator which is not available thruout the universe. If Kc is not fully iterable, one can't use it to isolate K, and in such a situation usually there is no inner model which has enough of the relevant properties that it might be called "the core model." See e.g. the first two pages of 
 <https://ivv5hpp.uni-muenster.de/u/rds/K.pdf> for a list of such properties.
 (There are exceptions to this, though, see e.g. <https://ivv5hpp.uni-muenster.de/u/rds/CMPW.pdf> and also <https://ivv5hpp.uni-muenster.de/u/rds/RalfGrigor_version_Sept11_2017.pdf> .)


