What are the Possible Large Cardinals of $L[X]$?

Question

I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the motivation for these constructions since they are much more complicated than the constructible universe $L$, or models like $L[X]$ built by adding a predicate to the construction of $L$. I am wondering, what makes a naive approach like taking $L[X]$ for the right sets fail for larger cardinals? Are there some results that pinpoint at what large cardinal strength more complicated constructions than $L[X]$ are needed? Thank you.

Answer 1

If $X$ is a set, then $L[X]$ does not have strong cardinals. In fact, $X^\sharp$ does not belong to $L[X]$, so any assumption that implies that $V$ is closed under sharps fails in these models.

Now, this is perhaps not the right example, as strong cardinals or supercompact cardinals essentially require a proper class of "witnesses". If you just want "local" versions, for example, having your model satisfy that there is an inaccessible $\kappa$ with $V_\kappa$ a model of "there is a proper class of strong cardinals" or more (say, having Woodin cardinals in your model), then models $L[X]$ are fine. In fact, contrary to your claim, the Mitchell-Steel models for Woodin cardinals are models of the form $L[X]$ for an appropriate set $X$.

So perhaps we need to refine your question to something much more specific, such as why we need complicated sequences of extenders (or measures) rather than just having $X$ be, say, an ultrafilter. This is easy to see when we try to model a degree of supercompactness, because nontrivial ultrafilters on $\mathcal P_\kappa(\lambda)$ concentrate on non-trivial sets (rather than sets of ordinals), so adding such an ultrafilter as a predicate $X$ ends up not doing much, since $L[X]$ cannot see any interesting measure $1$ sets, so $L[X]$ reduces to $L$ in this case. This has been known forever, it was noticed quickly after the $L[\mu]$ models for measurability began to be investigated.

Again, the above may not be the best example, since there are additional obstacles to developing fine-structure theory even at the level of $\kappa^+$-strong compactness, so we do not have "Mitchell-Steel" models for this assumption anyway.

Now, the complicated $L[\mathcal E]$ models are complicated because they are trying to do more than just modeling the large cardinal assumptions we are after. For example, to model that $\delta$ is Woodin, we need many "local" strong cardinals below $\delta$, so we need to add many ultrafilters to the model; this already indicates our constructions will have $\mathcal E$ be not a single measure, but rather a sequence of measures.

But the $L[\mathcal E]$ models do more: We want to add these ultrafilters in increasing order of "strength", which is measured by several parameters, such as their Mitchell order. This requires that we be able to organize the ultrafilters in some ordered fashion, avoiding clashes, so that if at some stage $\alpha$ we are to add a measure, there is exactly one candidate measure to add. This is useful to establish basic results such as condensation, the higher order versions of the result that elementary substructures of initial segments of $L$ are again initial segments of $L$.

We also want our models to be as canonical as possible, so in particular we must avoid coding sets by accident, which could happen if we are not careful on what measures we put in $\mathcal E$. There are additional requirements, all meant to ensure that the models we obtain can be compared with one another, carry no unwanted information, and add strength "from the ground up". To formalize this carefully ends up requiring some non-trivial amount of fine structure, and the verification that these fine structural assumptions indeed allow these constructions to succeed ends up being a careful inductive argument. This is the real source of the difficulties and technicalities associated with the theory of these models.

I suggest that you read the initial sections of the paper by Steel for the Handbook of set theory. They do a good job of spelling out some of these concerns and of describing how precisely the predicate $X$ that are used end up coding the ultrafilters witnessing the large cardinal assumptions one is after.

It may actually be useful to read first the appropriate sections in Zeman's book. The points above are illustrated there carefully, and one avoids the additional mathematical complications that come from iteration trees, which are inherent to the theory of Woodin cardinals. Truth is, when one first encounters iteration trees, they may appear rather complicated. It is natural that something complicated is needed, since we want to produce models that can code complicated sets of reals, and the complexity of the possible sets of reals of these $L[\mathcal E]$ models is closely tied up to the complexity of the comparison process. (This is nicely explained at the beginning of the Martin-Steel paper on Iteration trees.)

What may not seem so natural is the solution of using iteration trees rather than other coding devices when indexing the ultrafilters in the sequences. In fact, other approaches were suggested, by Baldwin and others, more or less generalizing directly the theory we get at the level of models for $o(\kappa)=\kappa^{++}$. But these suggestions were the ones that ended up being rather complicated and led to the theory being stuck until Steel had the idea of iteration trees. In fact, the theory was stuck well before the level of Woodin cardinals. Thanks to the development of the Mitchell-Steel machinery, these prior levels could then be developed as well.