For over a century, the Danish thinker Søren Kierkegaard has been at the center of a number of important discussions, concerning not only philosophy and theology, but also, more recently, fields such as social thought, psychology, and contemporary aesthetics, especially literary theory. Despite his relatively short life, Kierkegaard was an extraordinarily prolific writer, as attested to by the 26-volume Princeton University Press edition of all of his published writings. But Kierkegaard left behind nearly as much unpublished writing, most of which (...) consists of what are called his "journals and notebooks." Kierkegaard has long been recognized as one of history's great journal keepers, but only rather small portions of his journals and notebooks are what we usually understand by the term “diaries.” By far the greater part of Kierkegaard’s journals and notebooks consists of reflections on a myriad of subjects—philosophical, religious, political, personal. Studying his journals and notebooks takes us into his workshop, where we can see his entire universe of thought. We can witness the genesis of his published works, to be sure—but we can also see whole galaxies of concepts, new insights, and fragments, large and small, of partially completed but unpublished works. Kierkegaard’s Journals and Notebooks enables us to see the thinker in dialogue with his times and with himself. Kierkegaard wrote his journals in a two-column format, one for his initial entries and the second for the extensive marginal comments that he added later. This edition of the journals reproduces this format, includes several photographs of original manuscript pages, and contains extensive scholarly commentary on the various entries and on the history of the manuscripts being reproduced. Volume 9 of this 11-volume series includes five of Kierkegaard’s important “NB” journals, which span from June 1852 to August 1854. This period was marked by Kierkegaard’s increasing preoccupation with what he saw as an unbridgeable gulf in Christianity—between the absolute ideal of the religion of the New Testament and the official, state-sanctioned culture of “Christendom,” which, embodied by the Danish People’s Church, Kierkegaard rejected with increasing vehemence. Crucially, Kierkegaard’s nemesis, Bishop Jakob Peter Mynster, died during this period and, in the months following, Kierkegaard can be seen moving inexorably toward the famous “attack on Christendom” with which he ended his life. (shrink)

For over a century, the Danish thinker Søren Kierkegaard has been at the center of a number of important discussions, concerning not only philosophy and theology, but also, more recently, fields such as social thought, psychology, and contemporary aesthetics, especially literary theory. Despite his relatively short life, Kierkegaard was an extraordinarily prolific writer, as attested to by the 26-volume Princeton University Press edition of all of his published writings. But Kierkegaard left behind nearly as much unpublished writing, most of which (...) consists of what are called his "journals and notebooks." Kierkegaard has long been recognized as one of history's great journal keepers, but only rather small portions of his journals and notebooks are what we usually understand by the term "diaries." By far the greater part of Kierkegaard's journals and notebooks consists of reflections on a myriad of subjects--philosophical, religious, political, personal. Studying his journals and notebooks takes us into his workshop, where we can see his entire universe of thought. We can witness the genesis of his published works, to be sure--but we can also see whole galaxies of concepts, new insights, and fragments, large and small, of partially completed but unpublished works. Kierkegaard's Journals and Notebooks enables us to see the thinker in dialogue with his times and with himself. Volume 4 of this 11-volume series includes the first five of Kierkegaard's well-known "NB" journals, which contain, in addition to a great many reflections on his own life, a wealth of thoughts on theological matters, as well as on Kierkegaard's times, including political developments and the daily press. Kierkegaard wrote his journals in a two-column format, one for his initial entries and the second for the extensive marginal comments that he added later. This edition of the journals reproduces this format, includes several photographs of original manuscript pages, and contains extensive scholarly commentary on the various entries and on the history of the manuscripts being reproduced. (shrink)

For over a century, the Danish thinker Søren Kierkegaard has been at the center of a number of important discussions, concerning not only philosophy and theology, but also, more recently, fields such as social thought, psychology, and contemporary aesthetics, especially literary theory. Despite his relatively short life, Kierkegaard was an extraordinarily prolific writer, as attested to by the 26-volume Princeton University Press edition of all of his published writings. But Kierkegaard left behind nearly as much unpublished writing, most of which (...) consists of what are called his "journals and notebooks." Kierkegaard has long been recognized as one of history's great journal keepers, but only rather small portions of his journals and notebooks are what we usually understand by the term "diaries." By far the greater part of Kierkegaard's journals and notebooks consists of reflections on a myriad of subjects--philosophical, religious, political, personal. Studying his journals and notebooks takes us into his workshop, where we can see his entire universe of thought. We can witness the genesis of his published works, to be sure--but we can also see whole galaxies of concepts, new insights, and fragments, large and small, of partially completed but unpublished works. Kierkegaard's Journals and Notebooks enables us to see the thinker in dialogue with his times and with himself. Volume 5 of this 11-volume series includes five of Kierkegaard's important "NB" journals, covering the months from summer 1848 through early May 1849. This was a turbulent period both in the history of Denmark--which was experiencing the immediate aftermath of revolution and the fall of absolutism, a continuing war with the German states, and the replacement of the State Church with the Danish People's Church--and for Kierkegaard personally. The journals in the present volume include Kierkegaard's reactions to the political upheaval, a retrospective account of his audiences with King Christian VIII, deliberations about publishing an autobiographical explanation of his writings, and an increasingly harsh critique of the Danish Church. These journals also reflect Kierkegaard's deep concern over his collision with the satirical journal Corsair, an experience that helped radicalize his view of "essential Christianity" and caused him to ponder the meaning of martyrdom. Kierkegaard wrote his journals in a two-column format, one for his initial entries and the second for the extensive marginal comments that he added later. This edition of the journals reproduces this format, includes several photographs of original manuscript pages, and contains extensive scholarly commentary on the various entries and on the history of the manuscripts being reproduced. (shrink)

For over a century, the Danish thinker Søren Kierkegaard has been at the center of a number of important discussions, concerning not only philosophy and theology, but also, more recently, fields such as social thought, psychology, and contemporary aesthetics, especially literary theory. Despite his relatively short life, Kierkegaard was an extraordinarily prolific writer, as attested to by the 26-volume Princeton University Press edition of all of his published writings. But Kierkegaard left behind nearly as much unpublished writing, most of which (...) consists of what are called his "journals and notebooks." Kierkegaard has long been recognized as one of history's great journal keepers, but only rather small portions of his journals and notebooks are what we usually understand by the term "diaries." By far the greater part of Kierkegaard's journals and notebooks consists of reflections on a myriad of subjects--philosophical, religious, political, personal. Studying his journals and notebooks takes us into his workshop, where we can see his entire universe of thought. We can witness the genesis of his published works, to be sure--but we can also see whole galaxies of concepts, new insights, and fragments, large and small, of partially completed but unpublished works. Kierkegaard's Journals and Notebooks enables us to see the thinker in dialogue with his times and with himself. Volume 6 of this 11-volume series includes four of Kierkegaard's important "NB" journals, covering the months from early May 1849 to the beginning of 1850. At this time Denmark was coming to terms with the 1848 revolution that had replaced absolutism with popular sovereignty, while the war with the German states continued, and the country pondered exactly what replacing the old State Church with the Danish People's Church would mean. In these journals Kierkegaard reflects at length on political and, especially, on ecclesiastical developments. His brooding over the ongoing effects of his fight with the satirical journal Corsair continues, and he also examines and re-examines the broader personal and religious significance of his broken engagement with Regine Olsen. These journals also contain reflections by Kierkegaard on a number of his most important works, including the two works written under his "new" pseudonym Anti-Climacus and his various attempts at autobiographical explanations of his work. And, all the while, the drumbeat of his radical critique of "Christendom" continues and escalates. Kierkegaard wrote his journals in a two-column format, one for his initial entries and the second for the extensive marginal comments that he added later. This edition of the journals reproduces this format, includes several photographs of original manuscript pages, and contains extensive scholarly commentary on the various entries and on the history of the manuscripts being reproduced. (shrink)

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, (...) incompleteness, and set theory--that give rise again and again to philosophical considerations. (shrink)

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal κ is supercompact if and only if every Π11 sentence true in a structure M (of any size) containing κ in (...) a language of size less than κ is also true in a substructure m≺M of size less than κ with m∩κ∈κ. (shrink)

The Lévy-Solovay Theorem [8] 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.

With this first part of the eleventh volume, Bruce Kirmmse et al.’s monumental task of translating Søren Kierkegaard’s journals and notebooks begins to draw to a close. The journals and notebooks t...

A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise definable. (...) Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. (shrink)

We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...) model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it. (shrink)

We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by (...) adding the singleton operator, are foundationally robust. (shrink)

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 κ -directed closed posets which preserve a supercompact cardinal κ 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. (shrink)

I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game‐theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of n letters, including infinite words (...) or even uncountable words, the codebreaker can nevertheless always win in n steps. Meanwhile, the mastermind number, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length ω over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of, for it is provably equal to the eventually different number, which is the same as the covering number of the meager ideal. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum. (shrink)

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (shrink)

We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only (...) the sets that are definable over what has been built so far, but also those that are algebraic over the existing structure. While we know that $\mathrm{Imp}$ can differ from $L$, the subtler properties of this new inner model are just now coming to light. Many questions remain open. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of (...) Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ]. (shrink)

The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.

A textbook for students who are learning how to write a mathematical proof, a validation of the truth of a mathematical statement.

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one set-forcing ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed (...) from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions. (shrink)

We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo–Fraenkel set theory $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\mathrm {ZFC}^{-}$ that are bi-interpretable, but not isomorphic—even $\langle H_{\omega _1},\in (...) \rangle $ and $ \langle H_{\omega _2},\in \rangle $ can be bi-interpretable—and there are distinct bi-interpretable theories extending $\mathrm {ZFC}^{-}$. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. (shrink)

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 κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...) add a Cohen subset to κ, the forcing to shoot a club Cκ avoiding 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. (shrink)

The infinite time Turing machine analogue of Post's problem, the question whether there are semi-decidable supertask degrees between 0 and the supertask jump 0∇, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0∇, but in the context of sets of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ a kind of transfinite-injury construction which generalizes canonically to oracles.

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) (...) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

If the Wholeness Axiom wa $_0$ is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa $_0$ is finitely axiomatizable.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

After small forcing, any $ -closed forcing will destroy the supercompactness and even the strong compactness of κ.

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

If G is a centreless group, then τ denotes the height of the automorphism tower of G. We prove that it is consistent that for every cardinal λ and every ordinal α<λ, there exists a centreless group G such that τ=α; and if β is any ordinal such that 1β<λ, then there exists a notion of forcing , which preserves cofinalities and cardinalities, such that τ=β in the corresponding generic extension.

Can a supercompact cardinal κ 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 κ, 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 halting problem for Turing machines is decidable on a set of asymptotic probability one. The proof is sensitive to the particular computational models.

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N (...) or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (shrink)

We define a potentialist system of ‐structures, i.e., a collection of possible worlds in the language of connected by a binary accessibility relation, achieving a potentialist account of the full background set‐theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the (...) modal theory. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory, both for assertions in the language of and for assertions in the full potentialist language. (shrink)

Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all.

We investigate various strong notions of rigidity for Souslin trees, separating them under ♢ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ♢ that there is a group whose automorphism tower is highly malleable by forcing.

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 Theorems 1 and 2 of [5], due to the first author. (shrink)

Superstrong cardinals 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, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q} \in V_\theta}$$\end{document}, the cardinal κ will exhibit none of the large cardinal properties with target θ or larger. (shrink)