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The answer to the "More precisely ..." question is that yes, it may be the case that $L$ has a single inaccessible cardinal $\kappa$ and that for no larger $\beta >\kappa$ is there any $\alpha$ with $ …

I think for the Theorem one can get by with a subcompact cardinal, which is weaker than a $\kappa^+$-supercompact. ( $\kappa$ is subcompact if for any $A\subseteq H_{\kappa^+}$ there are $j,\mu < \ka …

To see that the set of ineffables is stationary, let $C\subseteq \eta_\omega$ be any closed and unbounded set. It suffices to show that the good indiscernibles arising in the definition of $\omega$-Er …

It can also be the case that $I_A$ is a periodic (but club) subclass of $I$: by Jensen Coding one can define (necessarily by class forcing) reals $a\subset\omega$ with $0^\sharp \notin L[a]$ and that …

I think that in some ways you have answered your question yourself: we see that to prove properties about sets, say within the projective hierarchy, we need representations of those sets of reals as t …

Yes: an assumption like the one you quote for inaccessibles in L, namely $0^\sharp$. Instead you take a "mouse" i.e. an iterable structure, which has a measure of Mitchell order 1 as the topmost fina …

Addressing the first question: I should argue that Choice comes in almost at the beginning of the inner model project, if we regard proving Covering Lemmata as an integral part of that project: one de …

My reading of this question was different from Andreas', because Norman asked for order preserving maps of the indiscernibles to extend to embeddings $j:L[U]_\theta \rightarrow$ $L[U] _\theta$ i.e …

The answer to Q2 is 'No'. Suppose $j:L\rightarrow L$ is a non-trivial elementary embedding. We use the following fact: $\bullet$ $cp(j)$ (the first ordinal moved by $j$) is always a Silver indiscer …

A proper class of measurables more than suffices. It suffices for the generic absoluteness to have X-sharp exists for every set of ordinals X. Then the Martin-Solovay tree can be constructed through …

Mathias has a paper where he corrects the flaws that occur in Devlin's theory BS (= Basic Set Theory). The theory has to be only slightly strengthened to be correct. (It is more than sufficent to add …

There is a theorem of Gitik, Kanovei and Koepke that characterises the degrees of constructibility in $M[G]$ where $G$ is prikry generic over the model $M$: they are isomorphic to the $P(\omega)/Fin$ …

I agree with the last sentence of Bill Mitchell's comment. But here is something closer than the cardinals mentioned. In [1] the notion of "almost Ramsey" cardinal was coined. Such a cardinal $\kappa$ …

I think the short answer is that we are still a long way from a full core model for a supercompact cardinal. We have core models (in the sense of Steel's `Core Model Iterability Problem' Lecture Note …

One point to make is that no one embedding witnesses that a cardinal is strong (that is unless it is witnessing that $\kappa$ is something much stronger, like a supercompact). The definition requires …