Remarkable Laver functions

This is a talk at the CUNY Set Theory Seminar, February 27, 2015.

Laver famously defined and used a Laver function on a supercompact cardinal for the indestructibility argument showing that a supercompact cardinal can be made indestructible by all $\lt\kappa$-directed closed forcing [1]. Since then, Laver-like functions for various other large cardinals have played in a significant role in helping to lift embeddings in indestructibility arguments. They also possess inherent interest as guessing functions with strong affinity to the $\diamondsuit$ principle, that arise specifically in the context of large cardinals.

Suppose that $\kappa$ is some large cardinal that can be characterized by the existence of certain kinds of elementary embeddings. A Laver-like (partial) function $\ell:\kappa\to V_\kappa$ roughly has the property that for every set $a$ in the universe, there is some embedding $j$ of the type characterizing the cardinal such that $j(\ell)(\kappa)=a$. In this sense, a Laver-like function can ``guess" or anticipate any element of the universe as the image of $\kappa$ under some embedding $j$. Large cardinals that don't have embeddings whose targets include arbitrarily large segments of the universe can have appropriately defined Laver-like functions that anticipate only elements of some subset of $V$. If $\kappa$ is measurable, then there is an embedding $j:V\to M$ but $M$ is only guaranteed to include $H_{\kappa^+}$ of $V$. So it would only make sense for a Laver-like function for a measurable cardinal to anticipate elements of $H_{\kappa^+}$. On the other hand, because a supercompact cardinal has embeddings with targets that include arbitrarily large rank initial segments of $V$, its Laver function can anticipate any set in the universe.

Example 1: Suppose that $\kappa$ is supercompact. A Laver function $\ell:\kappa\to V_\kappa$ has the property that for every cardinal $\lambda$ and $a\in H_{\lambda^+}$, there is a $\lambda$-supercompactness embedding $j:V\to M$ (with $M^\lambda\subseteq M$) such that $j(\ell)(\kappa)=a$.

Example 2: Suppose that $\kappa$ is extendible. An extendible Laver function $\ell:\kappa\to V_\kappa$ has the property that for every cardinal $\alpha$ and $a\in V_\alpha$, there is $j:V_\alpha\to V_\beta$ with $j(l)(\kappa)=a$.

Example 3: Suppose that $\kappa$ is measurable. A measurable Laver function $\ell:\kappa\to V_\kappa$ has the property that for every $a\in H_{\kappa^+}$, there is $j:V\to M$ such that $j(\ell)(\kappa)=a$.

Example 4: Suppose that $\kappa$ is strongly unfoldable. A strongly unfoldable Laver function $\ell:\kappa\to V_\kappa$ has the property that for every $a\in H_{\theta^+}$, there is a $\theta$-strong unfoldability embedding $j:M\to N$ (with $V_\theta\subseteq N$) such that $j(\ell)(\kappa)=a$.

Although, the existence of Laver-like functions can be forced for many large cardinals, only few large cardinals such as supercompact, strong, and extendible cardinals have them outright. A very general account of Laver-like functions for various large cardinals and where they exist or can be forced to exist appears in [2]. For instance, not every strongly unfoldable cardinal has a strongly unfoldable Laver function because it is consistent that $\diamondsuit_\kappa({\rm REG})$ fails at a strongly unfoldable cardinal [3], but every universe with a strongly unfoldable cardinal has a forcing extension in which it has a strongly unfoldable Laver function.

I will show that remarkable cardinals have Laver-like functions, which we call remarkable Laver functions. To motivate the definition of remarkable cardinals, introduced by Ralf Schindler, let's consider the following theorem of Magidor giving an alternative characterization of supercompact cardinals in terms of small embeddings.

Theorem 1: (Magidor, [4]) A cardinal $\kappa$ is supercompact if and only if for every regular cardinal $\lambda>\kappa$, there is a regular cardinal $\overline \lambda<\kappa$ and $j:H_{\overline\lambda}\to H_\lambda$ with critical point $\gamma$ such that $j(\gamma)=\kappa$.

Remarkable cardinals are pseudo-supercompact in the sense that a cardinal is remarkable if the embeddings characterizing a supercompact cardinal exist, not necessarily in $V$, but instead somewhere in $V$'s generic multiverse.

Definition 1: A cardinal $\kappa$ is remarkable if for every regular cardinal $\lambda>\kappa$, there is a forcing extension $V[H]$ of $V$ having an embedding $j:H^V_{\overline\lambda}\to H^V_\lambda$ for some $V$-regular $\overline\lambda<\kappa$, with critical point $\gamma$ and $j(\gamma)=\kappa$.

It is not difficult to show that this definition is equivalent to the standard definition of remarkable cardinals [5] (see this post for a more extensive introduction to remarkable cardinals).

Observation 1: A cardinal $\kappa$ is remarkable if in the ${\rm Coll}(\omega,\lt\kappa)$ forcing extension $V[G]$, for every regular cardinal $\lambda>\kappa$, there is an embedding $j:H_{\overline\lambda}^V\to H_\lambda^V$ for some $V$-regular $\overline\lambda<\kappa$, with critical point $\gamma$ and $j(\gamma)=\kappa$.

In order to prove observation 1, we need a lemma about the absoluteness of existence of embeddings of countable models, which was discussed in this post and plays a central role in proving just about anything having to do with remarkable cardinals.

Lemma: (Absoluteness Lemma for countable embeddings) Suppose that $M$ and $N$ are transitive models of ${\rm ZFC}^-$ and $j:M\to N$ is an elementary embedding. If $W\subseteq V$ is a transitive (set or class) model of ${\rm ZFC}^-$ such that $M$ is countable in $W$ and $N\in W$, then $W$ has some elementary embedding $j^*:M\to N$. Moreover, if $\text{cp}(j)=\gamma$ and $j(\gamma)=\delta$, we can additionally assume that $\text{cp}(j^*)=\gamma$ and $j^*(\gamma)=\delta$. Additionally, we can assume that $j$ and $j^*$ agree on some fixed finite number of values.

Proof of observation 1: The existence of some $j:H_{\overline\lambda}^V\to H_\lambda^V$ for a sufficiently large $\lambda$ with $j(\gamma)=\kappa$, where $\gamma$ is the critical point of $j$, anywhere in the multiverse, implies that $\kappa$ is inaccessible. Suppose that $V[H]$ is some forcing extension of $V$ having an embedding $j:H_{\overline\lambda}^V\to H_{\lambda}^V$ with critical point $\gamma$, $j(\gamma)=\kappa$, and $\overline\lambda<\kappa$. Let's argue that every ${\rm Coll}(\omega,\lt\kappa)$-forcing extension of $V$ has such an embedding $j^*$ as well. If we suppose not, then there is some condition $p\in {\rm Coll}(\omega,\lt\kappa)$ forcing that there is no such embedding. Let $G$ be any $V[H]$-generic for ${\rm Coll}(\omega,\lt\kappa)$ containing $p$. Then $H_{\overline\lambda}^V$ is countable both in $V[H][G]$ and its submodel $V[G]$ (since $\kappa$ is inaccessible, meaning that $H_{\overline\lambda}^V$ has size less than $\kappa$). Thus, by the absoluteness lemma, there is $j^*:H_{\overline\lambda}^V\to H_\lambda^V$ in $V[G]$ such that $j^*(\gamma)=\kappa$. But this contradicts our assumption that $p\in G$ forces that no such embedding exists. $\square$

Now, let's define what is a remarkable Laver function.

Definition 2: Suppose that $\kappa$ is remarkable. We define that a function $\ell:\kappa\to V_\kappa$ is a remarkable Laver function if whenever $\lambda>\kappa$ is regular and $x\in H_\lambda$, then every ${\rm Coll}(\omega,\lt\kappa)$-forcing extension $V[G]$ has an embedding $j:H_{\overline\lambda}^V\to H_{\lambda}^V$, where $\overline\lambda<\kappa$ is $V$-regular, critical point is $\gamma$, and $j(\gamma)=\kappa$, such that

1. $\ell\upharpoonright\gamma+1\in H_{\overline\lambda}^V$,
2. $\gamma\in\text{dom}(\ell)$,
3. $j(\ell\upharpoonright\gamma+1)(\kappa)=x$.

Theorem 2: (Cheng, G., [bibcite key=chenggitman:IndestructibleRemarkableCardinals]) Every remarkable cardinal has a remarkable Laver function.

For indestructibility results, the value of a remarkable Laver function lies in the following theorem.

Theorem 3: (Cheng, G., [6]) Suppose that $\kappa$ is remarkable and $\ell$ is a remarkable Laver function. In a ${\rm Coll}(\omega,\lt\kappa)$-forcing extension $V[G]$, for every regular cardinal $\lambda>\kappa$, there is an embedding $j:H_{\overline\lambda}^V\to H_\lambda^V$, where $\overline\lambda<\kappa$ is $V$-regular, $\gamma$ is the critical point, and $j(\gamma)=\kappa$, such that

1. $(\gamma,\overline\lambda]\cap \text{dom}(\ell)=\emptyset$,
2. $\ell(\gamma)$ is defined,
3. $j(\ell\upharpoonright\gamma)=\ell$.

References

1. R. Laver, “Making the supercompactness of $κ$ indestructible under $κ$-directed closed forcing,” Israel J. Math., vol. 29, no. 4, pp. 385–388, 1978.
2. J. D. Hamkins, “A class of strong diamond principles,” Manuscript, 2002.
3. Dz̆amonja Mirna and J. D. Hamkins, “Diamond (on the regulars) can fail at any strongly unfoldable cardinal,” Ann. Pure Appl. Logic, vol. 144, no. 1-3, pp. 83–95, Dec. 2006.
4. M. Magidor, “On the role of supercompact and extendible cardinals in logic,” Israel J. Math., vol. 10, pp. 147–157, 1971.
5. R.-D. Schindler, “Proper forcing and remarkable cardinals,” Bull. Symbolic Logic, vol. 6, no. 2, pp. 176–184, 2000.
6. Y. Cheng and V. Gitman, “Indestructibility properties of remarkable cardinals,” Arch. Math. Logic, vol. 54, no. 7-8, pp. 961–984, 2015.