# Jacob Davis, PhD 2016, Carnegie Mellon University

Jacob Davis successfully defended his dissertation, “Universal Graphs at $\aleph_{\omega_1+1}$ and Set-theoretic Geology,” at Carnegie Mellon University on April 29, 2016, under the supervision of James Cummings. I was on the dissertation committee (participating via Google Hangouts), along with Ernest Schimmerling and Clinton Conley.

The thesis consisted of two main parts. In the first half, starting from a model of ZFC with a supercompact cardinal, Jacob constructed a model in which $2^{\aleph_{\omega_1}} = 2^{\aleph_{\omega_1+1}} = \aleph_{\omega_1+3}$ and in which there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$.  The same technique works with any uncountable cardinal in place of $\omega_1$.  In the second half, Jacob proved a variety of results in the area of set-theoretic geology, including several instances of the downward directed grounds hypothesis, including an analysis of the chain condition of the resulting ground models.

# Large cardinals need not be large in HOD

• Y. Cheng, S. Friedman, and J. D. Hamkins, “Large cardinals need not be large in HOD,” Annals of Pure and Applied Logic, vol. 166, iss. 11, pp. 1186-1198, 2015.
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Abstract. We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal $\kappa$ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.

In this article, we prove that large cardinals need not generally exhibit their large cardinal nature in HOD, the inner model of hereditarily ordinal-definable sets, and there can be a divergence in strength between the large cardinals of the ambient set-theoretic universe $V$ and those of HOD. Our general theme concerns the questions:

Questions.

1. To what extent must a large cardinal in $V$ exhibit its large cardinal properties in HOD?

2. To what extent does the existence of large cardinals in $V$ imply the existence of large cardinals in HOD?

For large cardinal concepts beyond the weakest notions, we prove, the answers are generally negative. In Theorem 4, for example, we construct a model with a supercompact cardinal that is not weakly compact in HOD, and Theorem 9 extends this to a proper class of supercompact cardinals, none of which is weakly compact in HOD, thereby providing some strongly negative instances of (1). The same model has a proper class of supercompact cardinals, but no supercompact cardinals in HOD, providing a negative instance of (2). The natural common strengthening of these situations would be a model with a proper class of supercompact cardinals, but no weakly compact cardinals in HOD. We were not able to arrange that situation, however, and furthermore it would be ruled out by Conjecture 13, an intriguing positive instance of (2) recently proposed by W. Hugh Woodin, namely, that if there is a supercompact cardinal, then there is a measurable cardinal in HOD. Many other natural possibilities, such as a proper class of measurable cardinals with no weakly compact cardinals in HOD, remain as open questions.

# Large cardinals need not be large in HOD, International Workshop on Set Theory, CIRM, Luminy, September 2014

I shall speak at the 13th International Workshop on Set Theory, held at the CIRM Centre International de Rencontres Mathématiques in Luminy near Marseille, France, September 29 to October 3, 2014.

Abstract.  I shall prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals. There are many open questions.

This talk will include joint work with Cheng Yong and Sy-David Friedman.

# Uniform $({\lt}\theta)$-supercompactness is equivalent to a coherent system of normal fine measures

$\newcommand\image{\mathrel{{}^{\prime\prime}}}$This post answers a question that had come up some time ago with Arthur Apter, and more recently with Philipp Schlicht and Arthur Apter.

Definition. A cardinal $\kappa$ is uniformly ${\lt}\theta$-supercompact if there is an embedding $j:V\to M$ having critical point $\kappa$, with $j(\kappa)>\theta$ and $M^{\lt\theta}\subset M$.

(Note:  This is typically stronger than merely asserting that $\kappa$ is $\gamma$-supercompact for every $\gamma<\theta$, a property which is commonly denoted ${\lt}\theta$-supercompact, so I use the adjective “uniformly” to highlight the distinction.)

Two easy observations are in order.  First, if $\theta$ is singular, then $\kappa$ is uniformly ${\lt}\theta$-supercompact if and only if $\kappa$ is $\theta$-supercompact, since the embedding $j:V\to M$ will have $j\image\lambda\in M$ for every $\lambda<\theta$, and we may assemble $j\image\theta$ from this inside $M$, using a sequence of length $\text{cof}(\theta)$. Second, in the successor case, $\kappa$ is uniformly ${\lt}\lambda^+$-supercompact if and only if $\kappa$ is $\lambda$-supercompact, since if $j:V\to M$ has $M^\lambda\subset M$, then it also has $M^{{\lt}\lambda^+}\subset M$. So we are mainly interested in the concept of uniform ${\lt}\theta$-supercompactness when $\theta$ is weakly inaccessible.

Definition. Let us say of a cardinal $\kappa$ that $\langle\mu_\lambda\mid\lambda<\theta\rangle$ is a coherent $\theta$-system of normal fine measures, if each $\mu_\lambda$ is a normal fine measure on $P_\kappa\lambda$, which cohere in the sense that if $\lambda<\delta<\theta$, then $\mu_\lambda\leq_{RK}\mu_\delta$, and more specifically $X\in\mu_\lambda$ if and only if $\{ \sigma\in P_\kappa\delta\mid \sigma\cap\lambda\in X\}\in\mu_\delta$.  In other words, $\mu_\lambda=f\ast\mu_\delta$, where $f:P_\kappa\delta\to P_\kappa\lambda$ is the function that chops off at $\lambda$, so that $f:\sigma\mapsto \sigma\cap\lambda$.

Theorem.  The following are equivalent, for any regular cardinals $\kappa\leq\theta$.

1. The cardinal $\kappa$ is uniformly ${\lt}\theta$-supercompact.

2. There is a coherent $\theta$-system of normal fine measures for $\kappa$.

Proof. The forward implication is easy, since if $j:V\to M$ has $M^{{\lt}\theta}\subset M$, then we may let $\mu_\lambda$ be the normal fine measure on $P_\kappa\lambda$ generated by $j\image\lambda$ as a seed, so that $X\in\mu_\lambda\iff j\image\lambda\in j(X)$.  Since the seeds $j\image\lambda$ cohere as initial segments, it follows that $\mu_\lambda\leq_{RK}\mu_\delta$ in the desired manner whenever $\lambda\lt\delta<\theta$.

Conversely, fix a coherent system $\langle\mu_\lambda\mid\lambda<\theta\rangle$ of normal fine measures. Let $j_\lambda:V\to M_\lambda$ be the ultrapower by $\mu_\lambda$. Every element of $M_\lambda$ has the form $j_\lambda(f)(j\image\lambda)$.  Because of coherence, we have an elementary embedding $k_{\lambda,\delta}:M_\lambda\to M_\delta$ defined by $$k_{\lambda,\delta}: j_\lambda(f)(j\image\lambda)\mapsto j_\delta(f)(j\image\lambda).$$ It is not difficult to check that these embeddings altogether form a commutative diagram, and so we may let $j:V\to M$ be the direct limit of the system, with corresponding embeddings $k_{\lambda,\theta}:M_\lambda\to M$.  The critical point of $k_{\lambda,\delta}$ and hence also $k_{\lambda,\theta}$ is larger than $\lambda$.  This embedding has critical point $\kappa$, and I claim that $M^{\lt\theta}\subset M$. To see this, suppose that $z_\alpha\in M$ for each $\alpha<\beta$ where $\beta<\theta$.  So $z_\alpha=k_{\lambda_\alpha,\theta}(z_\alpha^*)$ for some $z_\alpha^*\in M_{\lambda_\alpha}$. Since $\theta$ is regular, we may find $\lambda<\theta$ with $\lambda_\alpha\leq\lambda$ for all $\alpha<\beta$ and also $\beta\leq\lambda$, and so without loss we may assume $\lambda_\alpha=\lambda$ for all $\alpha<\beta$. Since $M_\lambda$ is closed under $\lambda$-sequences, it follows that $\vec z^*=\langle z_\alpha^*\mid\alpha<\beta\rangle\in M_\lambda$.  Applying $k_{\lambda,\theta}$ to $\vec z^*$ gives precisely the desired sequence $\vec z=\langle z_\alpha\mid\alpha<\beta\rangle$ inside $M$, showing this instance of $M^{{\lt}\theta}\subset M$. QED

The theorem does not extend to singular $\theta$.

Theorem.  If $\kappa$ is $\theta$-supercompact for a singular strong limit cardinal $\theta$ above $\kappa$, then there is a transitive inner model in which $\kappa$ has a coherent system $\langle\mu_\lambda\mid\lambda<\theta\rangle$  of normal fine measures, but $\kappa$ is not uniformly ${\lt}\theta$-supercompact.

Thus, the equivalence of the first theorem does not hold generally for singular $\theta$.

Proof.  Suppose that $\kappa$ is $\theta$-supercompact, where $\theta$ is a singular strong limit cardinal. Let $j:V\to M$ be a witnessing embedding, for which $\kappa$ is not $\theta$-supercompact in $M$ (use a Mitchell-minimal measure).  Since $\theta$ is singular, this means by the observation after the definition above that $\kappa$ is not uniformly ${\lt}\theta$-supercompact in $M$. But meanwhile, $\kappa$ does have a coherent system of normal fine ultrafilters in $M$, since the measures defined by $X\in\mu_\lambda\iff j\image\lambda\in j(X)$ form a coherent system just as in the theorem, and the sequence $\langle\mu_\lambda\mid\lambda<\theta\rangle$ is in $M$ by $\theta$-closure. QED

The point is that in the singular case, the argument shows only that the direct limit is ${\lt}\text{cof}(\theta)$-closed, which is not the same as ${\lt}\theta$-closed when $\theta$ is singular.

The example of singular $\theta$ also shows that $\kappa$ can be ${\lt}\theta$-supercompact without being uniformly ${\lt}\theta$-supercompact, since the latter would imply full $\theta$-supercompactness, when $\theta$ is singular, but the former does not. The same kind of reasoning separates uniform from non-uniform ${\lt}\theta$-supercompactness, even when $\theta$ is regular.

Theorem. If $\kappa$ is uniformly ${\lt}\theta$-supercompact for an inaccessible cardinal $\theta$, then there is a transitive inner model in which $\kappa$ is ${\lt}\theta$-supercompact, but not uniformly ${\lt}\theta$-supercompact.

Proof. Suppose that $\kappa$ is uniformly ${\lt}\theta$-supercompact, witnessed by embedding $j:V\to M$, with $M^{\lt\theta}\subset M$, and furthermore assume that $j(\kappa)$ is as small as possible among all such embeddings. It follows that there can be no coherent $\theta$-system of normal fine measures for $\kappa$ inside $M$, for if there were, the direct limit of the associated embedding would send $\kappa$ below $j(\kappa)$, which from the perspective of $M$ is a measurable cardinal far above $\kappa$ and $\theta$. But meanwhile, $\kappa$ is $\beta$-supercompact in $M$ for every $\beta<\theta$. Thus, $\kappa$ is ${\lt}\theta$-supercompact in $M$, but not uniformly ${\lt}\theta$-supercompact, and so the notions do not coincide. QED

Meanwhile, if $\theta$ is weakly compact, then the two notions do coincide. That is, if $\kappa$ is ${\lt}\theta$-supercompact (not necessarily uniformly), and $\theta$ is weakly compact, then in fact $\kappa$ is uniformly ${\lt}\theta$-supercompact, since one may consider a model $M$ of size $\theta$ with $\theta\in M$ and $V_\theta\subset M$, and apply a weak compactness embedding $j:M\to N$. The point is that in $N$, we get that $\kappa$ is actually $\theta$-supercompact in $N$, which provides a uniform sequence of measures below $\theta$.

# Large cardinal indestructibility: two slick new proofs of prior results

$\newcommand\HOD{\text{HOD}}$

I’ve recently found two slick new proofs of some of my prior results on indestructibility, using the idea of an observation of Arthur Apter’s.  What he had noted is:

Observation. (Apter [1])  If $\kappa$ is a Laver indestructible supercompact cardinal, then $V_\kappa\subset\HOD$.  Indeed, $V_\kappa$ satisfies the continuum coding axiom CCA.

Proof. The continuum coding axiom asserts that every set of ordinals is coded into the GCH pattern (it follows that they are each coded unboundedly often). If $x\subset\kappa$ is any bounded set of ordinals, then let $\mathbb{Q}$ be the forcing to code $x$ into the GCH pattern at regular cardinals directly above $\kappa$. This forcing is ${\lt}\kappa$-directed closed, and so by our assumption, $\kappa$ remains supercompact and in particular $\Sigma_2$-reflecting in the extension $V[G]$. Since $x$ is coded into the GCH pattern of $V[G]$, it follows by reflection that $V_\kappa=V[G]_\kappa$ must also think that $x$ is coded, and so $V_\kappa\models\text{CCA}$. QED

First, what I noticed is that this immediately implies that small forcing ruins indestructibility:

Theorem. (Hamkins, Shelah [2], Hamkins [3]) After any nontrivial forcing of size less than $\kappa$, the cardinal $\kappa$ is no longer indestructibly supercompact, nor even indestructibly $\Sigma_2$-reflecting.

Proof.  Nontrivial small forcing $V[g]$ will add a new set of ordinals below $\kappa$, which will not be coded unboundedly often into the continuum function of $V[g]$, and so $V[g]_\kappa$ will not satisfy the CCA.  Hence, $\kappa$ will not be indestructibly $\Sigma_2$-reflecting there. QED

This argument can be seen as essentially related to Shelah’s 1998 argument, given in [2].

Second, I also noticed that a similar idea can be used to prove:

Theorem. (Bagaria, Hamkins, Tsaprounis, Usuba [4])  Superstrong and other large cardinals are never Laver indestructible.

Proof.  Suppose the superstrongness of $\kappa$ is indestructible. It follows by the observation that $V_\kappa$ satisfies the continuum coding axiom. Now force to add a $V$-generic Cohen subset $G\subset\kappa$.  If $\kappa$ were superstrong in $V[G]$, then there would be $j:V[G]\to M$ with $V[G]_{j(\kappa)}=M_{j(\kappa)}$. Since $G$ is not coded into the continuum function, $M_{j(\kappa)}$ does not satisfy the CCA.  This contradicts the elementarity $V_\kappa=V[G]_\kappa\prec M_{j(\kappa)}$. QED

The argument shows that even the $\Sigma_3$-extendibility of $\kappa$ is never Laver indestructible.

I would note, however, that the slick proof does not achieve the stronger result of [4], which is that superstrongness is never indestructible even by $\text{Add}(\kappa,1)$, and that after forcing to add a Cohen subset to $\kappa$ (among any of many other common forcing notions), the cardinal $\kappa$ is never $\Sigma_3$-extendible (and hence not superstrong, not weakly superstrong, and so on).  The slick proof above uses indestructibility by the coding forcing to get the CCA in $V_\kappa$, and it is not clear how one would argue that way to get these stronger results of [4].

[1] Arthur W. Apter and Shoshana Friedman. HOD-supercompactness, inestructibility, and level-by-level equivalence, to appear in Bulletin of the Polish Academy of Sciences (Mathematics).

[2] Joel David Hamkins, Saharon Shelah, Superdestructibility: A Dual to Laver’s Indestructibility,  J. Symbolic Logic, Volume 63, Issue 2 (1998), 549-554.

[3] Joel David Hamkins, Small forcing makes any cardinal superdestructible, J. Symbolic Logic, 63 (1998).

[4] Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba, Superstrong and other large cardinals are never Laver indestructible, to appear in the Archive of Math Logic (special issue in memory of Richard Laver).

# Large cardinals need not be large in HOD, Rutgers logic seminar, April 2014

I shall speak at the Rutgers Logic Seminar on April 21, 2014, 5:00-6:20 pm, Room 705, Hill Center, Busch Campus, Rutgers University.

Abstract. I will show that large cardinals, such as measurable, strong and supercompact cardinals, need not exhibit their large cardinal nature in HOD.  Specifically, it is relatively consistent that a supercompact cardinal is not weakly compact in HOD, and one may construct models with a proper class of supercompact cardinals, none of them weakly compact in HOD.  This is current joint work with Cheng Yong.

Article

# Superstrong and other large cardinals are never Laver indestructible, ASL 2014, Boulder, May 2014

This will be an invited talk at the ASL 2014 North American Annual Meeting (May 19-22, 2014) in the special session Set Theory in Honor of Rich Laver, organized by Bill Mitchell and Jean Larson.

Abstract.  The large cardinal indestructibility phenomenon, discovered by Richard Laver with his seminal result on supercompact cardinals, is by now often seen as pervasive in the large cardinal hierarchy. Nevertheless, a new never-indestrucible phenomenon has emerged.  Superstrong cardinals, for example, are never Laver indestructible.  Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals (all for $n\geq 3$) are never Laver indestructible.  The proof involves a detailed technical analysis of the complexity of the definition in Laver’s theorem on the definability of the ground model, thereby involving and extending results in set-theoretic geology.  This is joint work between myself and Joan Bagaria, Kostas Tasprounis and Toshimichi Usuba.

# Brent Cody

Brent Cody earned his Ph.D. under my supervision at the CUNY Graduate Center in June, 2012.  Brent’s dissertation work began with the question of finding the exact consistency strength of the GCH failing at a cardinal $\theta$, when $\kappa$ is $\theta$-supercompact.  The answer turned out to be a $\theta$-supercompact cardinal that was also $\theta^{++}$-tall.  After this, he quickly dispatched more general instances of what he termed the Levinski property for a variety of other large cardinals, advancing his work towards a general investigation of the Easton theorem phenomenon in the large cardinal context, which he is now undertaking.  Brent held a post-doctoral position at the Fields Institute in Toronto, afterwards taking up a position at the University of Prince Edward Island.  He is now at Virginia Commonwealth University.

Brent Cody

Brent Cody, “Some Results on Large Cardinals and the Continuum Function,” Ph.D. dissertation for The Graduate Center of the City University of New York, June, 2012.

Abstract.  Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function with $F”\delta\subseteq\delta$ and GCH holds, then there is a cofinality preserving forcing extension in which $2^\gamma= F(\gamma)$ for each regular cardinal $\gamma<\delta$, and in which $\delta$ remains Woodin.

I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$ is equiconsistent with the existence of a cardinal $\kappa$ that is $\lambda$-supercompact and $\lambda^{++}$-tall.

I generalize a theorem on measurable cardinals due to Levinski, which says that given a measurable cardinal, there is a forcing extension preserving the measurability of $\kappa$ in which $\kappa$ is the least regular cardinal at which GCH holds. Indeed, I show that Levinski’s result can be extended to many other large cardinal contexts. This work paves the way for many additional results, analogous to the results stated above for Woodin cardinals and partially supercompact cardinals.

# Exactly controlling the non-supercompact strongly compact cardinals

• A. W.~Apter and J. D. Hamkins, “Exactly controlling the non-supercompact strongly compact cardinals,” J.~Symbolic Logic, vol. 68, iss. 2, pp. 669-688, 2003.
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We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.

# Indestructibility and the level-by-level agreement between strong compactness and supercompactness

• A. W.~Apter and J. D. Hamkins, “Indestructibility and the level-by-level agreement between strong compactness and supercompactness,” J.~Symbolic Logic, vol. 67, iss. 2, pp. 820-840, 2002.
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Can a supercompact cardinal $\kappa$ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above $\kappa$, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.

# The lottery preparation

• J. D. Hamkins, “The lottery preparation,” Ann.~Pure Appl.~Logic, vol. 101, iss. 2-3, pp. 103-146, 2000.
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The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal $\kappa$, for example, becomes fully indestructible by $\kappa$-directed closed forcing; a strong cardinal $\kappa$ becomes indestructible by less-than-or-equal-$\kappa$-strategically closed forcing; and a strongly compact cardinal $\kappa$ becomes indestructible by, among others, the forcing to add a Cohen subset to $\kappa$, the forcing to shoot a club $C$ in $\kappa$ which avoids the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context.

# Gap forcing: generalizing the Lévy-Solovay theorem

• J. D. Hamkins, “Gap forcing: generalizing the Lévy-Solovay theorem,” Bulletin of Symbolic Logic, vol. 5, iss. 2, pp. 264-272, 1999.
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The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

# Superdestructibility: a dual to Laver's indestructibility

• J. D. Hamkins and S. Shelah, “Superdestructibility: a dual to Laver’s indestructibility,” J.~Symbolic Logic, vol. 63, iss. 2, pp. 549-554, 1998. ([HmSh:618])
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After small forcing, any $<\kappa$-closed forcing will destroy the supercompactness, even the strong compactness, of $\kappa$.

# Destruction or preservation as you like it

• J. D. Hamkins, “Destruction or preservation as you like it,” Annals of Pure and Applied Logic, vol. 91, iss. 2-3, pp. 191-229, 1998.
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primaryClass = {math.LO},
}

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of ${<}\kappa$-directed closed posets which preserve a supercompact cardinal $\kappa$ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible.

# Canonical seeds and Prikry trees

• J. D. Hamkins, “Canonical seeds and Prikry trees,” J.~Symbolic Logic, vol. 62, iss. 2, pp. 373-396, 1997.
@article {Hamkins97:Seeds,
AUTHOR = {Hamkins, Joel David},
TITLE = {Canonical seeds and {P}rikry trees},
JOURNAL = {J.~Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {62},
YEAR = {1997},
NUMBER = {2},
PAGES = {373--396},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E40 (03E05 03E55)},
MRNUMBER = {1464105 (98i:03070)},
MRREVIEWER = {Douglas R.~Burke},
DOI = {10.2307/2275538},
URL = {http://jdh.hamkins.org/seeds/},
}

Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.