# Joint Laver diamonds I

I firmly believe that when one is stuck on a research problem one should tell as many people as possible about it, because one of two things will happen: either someone will solve your problem (and you will have contributed to the store of mathematical knowledge) or you will frustrate all of your mathematician friends. Either of those is a good thing. For this reason I’ve decided to write a couple of posts sketching my current project and some of the points of frustration.

The fundamental idea arose from the following fact about Laver functions on a supercompact cardinal, which was given to me as an exercise at some point by Joel Hamkins:

Theorem. If $\kappa$ is supercompact then there are $2^\kappa$ (the maximal possible number) many Laver functions $\langle \ell_\alpha;\alpha<2^\kappa\rangle$ such that this sequence is jointly Laver, i.e. such that for any $\theta$ and any sequence $\langle x_\alpha;\alpha<2^\kappa\rangle$ of sets in $H_{\theta^+}$ there is a $\theta$-supercompactness embedding $j\colon V\to M$ with critical point $\kappa$ which satisfies $j(\ell_\alpha)(\kappa)=x_\alpha$ for all $\alpha$.

It is not difficult to see that this is true; furthermore, it will be interesting to give an alternative proof of the weaker claim that there is a jointly Laver sequence of length $\kappa$. This can be accomplished by simply coding everything appropriately. Specifically, start with a single Laver function $\ell\colon \kappa\to V_\kappa$ and let $\ell_\alpha(\xi)=\ell(\xi)(\alpha)$. Given a sequence $\vec{x}=\langle x_\alpha;\alpha<\kappa\rangle$ in $H_{\theta^+}$, we fix a $\theta$-supercompactness embedding $j$ such that $j(\ell)(\kappa)=\vec{x}$. It is then easy to check that $j(\ell_\alpha)(\kappa)=x_\alpha$ for all $\alpha$.
For the $2^\kappa$ length case, we reindex our sequences to use elements of $\mathcal{P}(\kappa)$ instead of $2^\kappa$. Still working with a given Laver function $\ell$ we define $\ell_A(\xi)=\ell(\xi)(A\cap \xi)$ for $A\in\mathcal{P}(\kappa)$. Given a sequence $\vec{x}=\langle x_A;A\in\mathcal{P}(\kappa)\rangle$ in $H_{\theta^+}$ we fix a $(2^\kappa + \theta)$-supercompactness embedding $j$ such that $j(\ell)(\kappa)=\vec{x}$ and check that this $j$ makes everything work as required. If we had $\theta<2^\kappa$ we should, at the end, also factor our a supercompactness embedding of the appropriate degree.

There are two things we should take away from this proof:

1. The short case was fairly easy, requiring merely some coding. This suggests that whenever we have any Laver function-like object on a cardinal $\kappa$ we should have $\kappa$ many joint such objects, whatever that might mean;
2. In the long case we seemingly only used the $(2^\kappa+\theta)$-supercompactness of $\kappa$. If we then consider only partially supercompact cardinals, this raises the question whether there is any strength in having a length $2^\kappa$ jointly Laver sequence or whether such things just always exist (provided there is an appropriate Laver function in the first place).

As alluded to in point 1, questions about jointly Laver sequences make sense whenever a Laver function-like object makes sense. This ties together nicely with the various Laver diamond principles, introduced for many large cardinals in an (as of yet unpublished) paper by Hamkins. Building on his work and also on the work of Apter-Cummings-Hamkins on the number of measures problem, some answers have been forthcoming.

Let me illustrate the main results about these joint Laver sequences in the case of measurable cardinals, where many of the interesting phenomena already occur. The supercompact case is fairly similar, with some complications.

To be concrete, if $\kappa$ is measurable we call a function $\ell\colon \kappa\to V_\kappa$Laver diamond (for measurability) if for every $x\in H_{\kappa^+}$ there is an elementary embedding $j\colon V\to M$ with critical point $\kappa$ such that $j(\ell)(\kappa)=x$. We call a sequence $\langle \ell_\alpha;\alpha<\beta\rangle$joint Laver diamond sequence (for measurability) if for every sequence $\langle x_\alpha;\alpha<\beta\rangle$ of sets in $H_{\kappa^+}$ there is an elementary embedding $j\colon V\to M$ with critical point $\kappa$ such that $j(\ell_\alpha)(\kappa)=x_\alpha$ for every $\alpha$.

Theorem. If $\kappa$ is measurable and has a Laver diamond then it has a joint Laver diamond sequence of length $\kappa$. In general, if $\kappa$ is measurable then there is a forcing extensions in which $\kappa$ remains measurable and has a joint Laver diamond sequence of length $\kappa$.

This is quite simple. If there is a Laver diamond for $\kappa$ then we can simply do the coding we did before and get a joint Laver diamond sequence. The point is that if there is no Laver diamond for $\kappa$ we can always force to add one. This can be done in one of several (nonequivalent) ways, e.g. by Woodin’s fast function forcing or by first doing a preparatory forcing and then adding a Cohen subset to $\kappa$.

Theorem. If $\kappa$ is measurable then there is a forcing extension in which $\kappa$ remains measurable and has a joint Laver diamond sequence of length $2^\kappa$.

This builds on the construction of adding a single Laver diamond. We first force the GCH to hold at $\kappa$ if necessary. We next prepare by doing a Silver-style iteration up to $\kappa$ where we add $\gamma^+$ many Cohen subsets to inaccessible $\gamma$. Finally we add $\kappa^+$ many Cohen subsets to $\kappa$. An argument as in the previous case shows that the Cohen subsets of $\kappa$ can be decoded into a joint Laver diamond sequence, and since GCH still holds at $\kappa$ at the end, there are $2^\kappa$ many. The crucial issue is showing that $\kappa$ remains measurable after this forcing. The usual lifting argument via master conditions doesn’t work since the generic is too big to be distilled down to a master condition. To solve this we use what has been called in the literature the “master filter argument”, where instead of building a single master condition we build a descending sequence of partial master conditions, which encode larger and larger pieces of the generic. The construction is quite sensitive and exploits, among other things, the continuity of the embedding $j$ at $\kappa^+$ (this becomes relevant in the supercompactness argument).

The fact that in the resulting model GCH holds in $\kappa$ is unavoidable without stronger hypotheses. The following question is still open.

Question. Given a model where $\kappa$ is measurable, GCH fails at $\kappa$ and $\kappa$ has a Laver diamond, is there a forcing extension preserving these facts where $\kappa$ has a joint Laver diamond sequence of length $2^\kappa$?

The final result on measurables is a separation of the conclusions of the previous two theorems. Therefore, while having a joint Laver diamond sequence of length $\kappa$ is no weaker in consistency strength than having a joint Laver diamond sequence of length $2^\kappa$, the outright implication still fails.

Theorem. If $\kappa$ is measurable then there is a forcing extension in which $\kappa$ remains measurable and has a joint Laver diamond sequence of length $\kappa$ but no joint Laver diamond sequence of length $\kappa^+$.

The key observation here is that, in order to have a joint Laver diamond sequence of length $\nu$, there must be at least $2^\nu$ many normal measures on $\kappa$, since every binary $\nu$-sequence must be guessed by some embedding and, of course, each embedding corresponds to a single sequence. The argument now proceeds by first forcing to add a Laver diamond to $\kappa$ as before and then using a result of Apter-Cummings-Hamkins by which we can force over a model with a measurable $\kappa$ preserving measurability but making $\kappa$ only carry $\kappa^+$ many normal measures. By our argument before $\kappa$ cannot possibly have a joint Laver diamond sequence of length greater than $\kappa$. It then remains to check that the single Laver diamond survived this final forcing and this gives us a joint Laver diamond sequence of length $\kappa$ as in our first theorem above.

The main gap in these results concerns the lack of control over $2^\kappa$. One would like to be able to push $2^\kappa$ high and still talk about joint Laver diamond sequences of intermediate length. Of course, this requires higher consistency strength than merely measurability, but I would guess that we get equiconsistency at that level again.

Next time (whenever that might be) I will discuss similar results on (partially) supercompact cardinals and perhaps some others (like weakly compact or strong or strongly unfoldable).

# Grounded Martin’s axiom

This is a short summary of some recent work on a principle I call the grounded Martin’s axiom. I gave a talk on this material in the CUNY Set Theory seminar a few days ago and a preprint will be available in the near future.

The grounded Martin’s axiom (or grMA) states that the universe $V$ is a ccc forcing extension of some ground model $W$ and that for any poset $P\in W$ which is ccc in $V$ and any collection $\mathcal{D}\in V$ of less than continuum many dense subsets of $P$ there is a $\mathcal{D}$-generic filter on $P$.

This concept appears naturally when one analyses the Solovay-Tennenbaum proof of the consistency of MA (with the continuum being a regular cardinal $\kappa$). There we iterate, in $\kappa$ many steps, through all the available ccc posets of size $<\kappa$ and use a suitable bookkeeping device to make sure that we have taken care of not only the posets in the ground model but also the posets that arise in all of the $\kappa$ many intermediate models as well. This bookkeeping device (basically a bijection between $\kappa$ and $\kappa\times\kappa$) will necessarily be wildly discontinuous and, in my opinion, distracts from the essence of the argument. Thus I have in the past suggested a reorganization of the proof which eliminates the need for (at least this part of the) bookkeeping by making the iteration slightly longer. Specifically, we construct a finite support iteration of length $\kappa^2$ as follows: starting in a suitable model (satisfying GCH or at least $2^{<\kappa}=\kappa$) we iterate the $\kappa$ many small ccc posets from this model, taking care to only take posets which remain ccc in the extension obtained so far; after the first $\kappa$ many steps we repeat the process, considering now the small ccc posets in this extension. And we do it again and again, $\kappa$ many times. The usual arguments show that what we get in the end is a model of MA and the continuum has size $\kappa$.

However, a new question now arises. Did we need to repeat this process $\kappa$ many times? Did we need to repeat it at all? Might we already have MA after the first $\kappa$ steps of the new iteration? The answer is no (assuming $\kappa>\omega_1$). To see why, notice that the forcing up to that point is an iteration of ground model posets, so it is basically a product. Since the forcing to add a single Cohen real will have inevitably appeared as a factor somewhere in this product, the model obtained is a Cohen extension of some intermediate model, but it is well known that MA fails in any Cohen extension where CH fails.

So MA fails in this model, but on the other hand, it looks perfectly crafted to satisfy grMA. Well, almost. What we have ensured by construction is that the restriction of grMA to posets of size less than $\kappa$ holds. The same issue arises in the usual MA argument and an easy Löwenheim-Skolem argument shows that there the two versions are equivalent. We cannot simply transpose the argument to the present context since the appropriate elementary substructure of the poset is now in the wrong model, but fortunately a modification of the argument gives the analogous result for grMA.

Having now what might be called a canonical model of grMA, we can also determine some cardinal characteristics in this model. Since grMA clearly implies MA(Cohen) we must have $\mathrm{cov}(\mathcal{B})=\mathfrak{c}$, but since, as before, the model is obtained by adding $\omega_1$ many Cohen reals to an intermediate extension, we can also conclude that $\mathrm{non}(\mathcal{B})=\omega_1$ in this model. These two equalities now resolve the whole of Cichoń’s diagram and also show that grMA is less rigid than MA with respect to some of the smaller cardinal characteristics.

Another noteworthy observation is that, while MA implies that the continuum is regular, grMA is consistent with a singular continuum. In particular, it is possible in a model of grMA to have $2^{<\mathfrak{c}}>\mathfrak{c}$, violating the generalized Luzin hypothesis. An interesting open question here is whether grMA implies that $2^{<\mathrm{cf}(\mathfrak{c})}=\mathfrak{c}$. While this equality holds in the canonical model, I do not know whether it holds in general.

The remainder of the current results on grMA concern its robustness under forcing. It is known that MA is destroyed by very mild forcing, adding either a Cohen or a random real (assuming CH fails). At the same time some fragments of MA are known to be preserved by such forcing. To determine the behaviour of grMA under such forcing, a variation of termspace forcing was utilized.

Termspace forcing (due to Laver and possibly independently Woodin and other people) is a construction for taking a two step forcing iteration $P\ast \dot{Q}$ and trying to approximate the poset named by $\dot{Q}$ by a poset in the ground model. This gives the poset $A(P,\dot{Q})$, consisting of $P$-names which are forced by every condition to be in $\dot{Q}$ and where $\tau$ extends $\sigma$ if this is forced by every condition. It can then be proved that forcing with $A(P,\dot{Q})$ adds a sort of doubly generic object for $\dot{Q}$. More precisely, forcing with $A(P,\dot{Q})$ gives a name which, when interpreted by any $V$-generic $G$ for $P$, names a $V[G]$-generic for $\dot{Q}^G$. In particular, the iteration $P\ast\dot{Q}$ embeds into the product $A(P,\dot{Q})\times P$.

The crucial issue, however, is that $A(P,\dot{Q})$ might not have any nontrivial chain conditions. This is clearly problematic for us, since we are dealing with an axiom that concerns only ccc posets. To fix this flaw we need to restrict the names we consider in the termspace forcing and for this purpose the notion of finite mixtures is introduced. A finite mixture is a $P$-name for an element of the ground model which is decided by some finite maximal antichain (the term finite mixture suggests that these names are obtained by applying the mixing lemma to finitely many check names). The subposet $A_{\mathrm{fin}}(P,\dot{Q})$ of $A(P,\dot{Q})$, consisting only of finite mixtures, has a much better chance of having a good chain condition. In particular, it can be seen that if $P$ is just the forcing to add a single Cohen real, then $A_{\mathrm{fin}}(P,Q)$ is Knaster if $Q$ is (here $Q$ is assumed to be in the ground model). This is the key step in showing that grMA is preserved by adding a single Cohen real (in fact it is preserved with respect to the same ground model). By slightly modifying the notion of a finite mixture to exploit the measure theory involved, a similar approach also shows that grMA is preserved by adding a random real (again, even with respect to the same ground model).

The question still remains whether grMA is preserved when adding more generic reals. For example, what happens if we add $\omega_1$ many Cohen reals? The methods used for a single real hinge on certain antichain refinement properties of the Cohen poset which are no longer there when adding more reals. Similar question can also be asked for random reals. In that case, at least, we do have an upper bound for preservation, as it is known that adding more than $\mathfrak{c}$ many random reals will destroy MA(Cohen) and thus also grMA, but nothing is known about adding a smaller number.