Let n ≥ 3 be an integer. We show that it is consistent that every σ1n-set of reals is universally Baire yet there is a projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk (...) that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)

This book introduces a new approach to the issue of radical scientific revolutions, or "paradigm-shifts," given prominence in the work of Thomas Kuhn. The book articulates a dynamical and historicized version of the conception of scientific a priori principles first developed by the philosopher Immanuel Kant. This approach defends the Enlightenment ideal of scientific objectivity and universality while simultaneously doing justice to the revolutionary changes within the sciences that have since undermined Kant's original defense of this ideal. Through a modified (...) Kantian approach to epistemology and philosophy of science, this book opposes both Quinean naturalistic holism and the post-Kuhnian conceptual relativism that has dominated recent literature in science studies. Focussing on the development of "scientific philosophy" from Kant to Rudolf Carnap, along with the parallel developments taking place in the sciences during the same period, the author articulates a new dynamical conception of relativized a priori principles. This idea applied within the physical sciences aims to show that rational intersubjective consensus is intricately preserved across radical scientific revolutions or "paradigm-shifts and how this is achieved. (shrink)

Preview: /Review: Randall Auxier, Eli Kramer, and Krzysztof Piotr Skowroński, eds. Rorty and Beyond, 314 pages./ Richard Rorty became a highly controversial figure, both within and without the ranks of academic philosophy upon publication of his 1979 book, Philosophy and the Mirror of Nature. The controversy over Rorty intensified with his writings over the remaining years of his life and beyond. Co-edited by Randall Auxier, Eli Kramer, and Krzysztof Piotr Skowroński and including sixteen original essays, this new book, Beyond Rorty (...) studies Rorty’s work with the passage of the years following his death to re-assess his significance and to consider ways in which philosophers might both learn from and move beyond his work. The volume originated in a conference held on Rorty’s work in 2013 at Opole University, Poland. It includes both papers presented at the conference together with additional submissions. The essays offer many perspectives on Rorty while seeking to move beyond his work in areas including metaphysics and epistemology, social philosophy, axiology, philosophy of religion, and more. The following review of Beyond Rorty discusses each of the essays in the volume together with the introductory essays by Auxier and Kramer to explore directions which contemporary philosophy might take to move beyond Rorty. (shrink)

Abstract In this paper I undertake an in-depth examination of an oft mentioned but rarely expounded upon state: suspended judgment. While traditional epistemology is sometimes characterized as presenting a “yes or no” picture of its central attitudes, in fact many of these epistemologists want to say that there is a third option: subjects can also suspend judgment. Discussions of suspension are mostly brief and have been less than clear on a number of issues, in particular whether this third option should (...) be thought of as an attitude or not. In this paper I argue that suspended judgment is (or at least involves) a genuine attitude. Content Type Journal Article Pages 1-17 DOI 10.1007/s11098-011-9753-y Authors Jane Friedman, St Catherine’s College, University of Oxford, Oxford, OX1 3UJ UK Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116. (shrink)

This book introduces a new approach to the issue of radical scientific revolutions, or "paradigm-shifts," given prominence in the work of Thomas Kuhn. The book articulates a dynamical and historicized version of the conception of scientific a priori principles first developed by the philosopher Immanuel Kant. This approach defends the Enlightenment ideal of scientific objectivity and universality while simultaneously doing justice to the revolutionary changes within the sciences that have since undermined Kant's original defense of this ideal. Through a modified (...) Kantian approach to epistemology and philosophy of science, this book opposes both Quinean naturalistic holism and the post-Kuhnian conceptual relativism that has dominated recent literature in science studies. Focussing on the development of "scientific philosophy" from Kant to Rudolf Carnap, along with the parallel developments taking place in the sciences during the same period, the author articulates a new dynamical conception of relativized a priori principles. This idea applied within the physical sciences aims to show that rational intersubjective consensus is intricately preserved across radical scientific revolutions or "paradigm-shifts and how this is achieved. (shrink)

Jensen's remarkable Coding Theorem asserts that the universe can be included in L[R] for some real R, via class forcing. The purpose of this article is to present a simpler proof of Jensen's theorem, obtained by implementing some changes first developed for the theory of Strong Coding. In particular, our proof avoids the split into cases, according to whether or not 0# exists in the ground model.

During an April 12, 1860, Senate debate, Senator Jefferson Davis spoke against a bill to fund Black education in Washington, D.C. Davis argued that the United States government was founded “by white men for white men” and “not for negroes.” According to Davis, the inequality of the white and black races was “stamped from the beginning.”Davis’s phrase forms the basis of Ibram X. Kendi’s 2016 National Book Award–winning study, Stamped from the Beginning: The Definitive History of Racist Ideas in America. (...) Kendi currently serves as the Andrew W. Mellon Professor in the Humanities and as the founding director of the Boston University Center for Antiracist Research. Stamped from the Beginning has won a large... (shrink)

By forcing over a model of with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H is a well-order of H definable over the structure H, by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in (...) addition to forcing a locally defined well-order of the universe, preserve many of the n-huge cardinals from the ground model. (shrink)

My focus in this work is on giving an analysis of Peirce's post-1895 remarks about realism and the realism/nominalism debate. I argue that there is a consistent position to be found in these writings, yet in order to understand his position we must look not only at Peirce's remarks on realism, but also to the various themes connected with his realism, viz. to his discussion of the categories, pragmatism, and opposing views. ;From Peirce's direct remarks on realism we learn that (...) there are two kinds of universals, viz., laws and possibilities , both of which are independent of mind. Peirce's claim that Firsts are real as possibilities, and Thirds are real as that to which Seconds conform, constitutes his universal-realism . Thirds, however, are laws, that which science seeks to discover. The subjunctive conditionals which express scientific theories are either true or false, and the entities to which they refer are real or figments. In this consists Peirce's scientific realism, or "scholastic realism". ;From Peirce's discussions of the categories and of nominalism, we learn that there are two kinds of qualities: embodied and unembodied. The latter are uninstantiated universals, the former are dispositions. Peirce's doctrine of pragmaticism is an analysis of embodied qualities, i.e., what it means to say that "an object possesses a character." I suggest that Peirce insists that uninstantiables as necessary for realism about laws because they ground the counterfactuals which law-statements support. ;Because Peirce has both uninstantiated universals, and because he also believes that there is a sense in which universals really are "in" things as parts, he is neither Transcendental nor Immanent Realist exclusively. (shrink)

The problem of Marxism is the problem of tyranny. The central argument against Marxism is an empirical one: the universally tyrannical nature of all hitherto existing Marxist regimes. Defenders of Marxism must continually defend themselves against the charge that Marxism, when it comes to power, increases the sum total of human misery by increasing political oppression. Marxists have answered in several ways. Some have argued that the social and economic benefits of Marxism outweigh the political misery it causes. Others have (...) argued that while tyranny might count against any particular regime, it is not intrinsic to Marxist regimes as such. Some have argued that tyranny is a transitional phase, necessary but impermanent. Finally, some Marxists have denied that the regimes they defend are tyrannical at all. (shrink)

Let n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n - 2 strong cardinals) that every $\Sigma_n^1-set$ of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses "David's trick" in the presence of inner models with strong cardinals.

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening (...) of Cohen's result:Theorem 1. There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals. (shrink)

ABSTRACTOn September 4, 2015, the Political Epistemology/ideas, Knowledge, and Politics section of the American Political Science Association sponsored a roundtable on ideational turns in the four subdisciplines of political science as part of its annual meetings. Chairing the roundtable was Jeffrey Friedman, Department of Government, University of Texas, Austin. The other participants were Jeffrey Checkel, Department of Political Science, Simon Fraser University; Matthias Matthijs, School of Advanced International Studies, Johns Hopkins University; and Rogers Smith, Department of Political Science, University (...) of Pennsylvania. We thank the participants for permission to republish their remarks, which they were offered the opportunity to edit after the fact. (shrink)

All axiomatizations in sections 1,2,4-8 are in the language L(Î,W) with just Î and the constant symbol W standing for a Subworld. Think of W as yesterday's world, and think of the quantifiers in the theory as ranging over today's world. The philosophy is that since the universe cannot be completed, every time we reflect on the universe and what we have reflected on previously, we obtain a larger universe.

Martin Buber was close to sociology and sociologists from his university years on and in 1938 was head of the new Department of Sociology at the Hebrew University of Jerusalem. Although influenced by Ferdinand Toennies, and George Simmel, he went beyond them in his philosophy of the “interhuman” from which standpoint he also criticized Max Scheler. Focal social concepts of Buber's are “the interhuman”_the dialogical relationship between persons that entails “inclusion,” or “imagining the real,” making present, and confirmation ; the (...) “essential We” or common cosmos that each helps to build by speech-with-meaning from his or her unique stance; the distinction between the “political principle” of government and the “social principle” of fellowship and social spontaneity; and the philosophy of community that led Buber to a federalistic socialism and the vision of restructuring society into a “community of communities”. (shrink)

Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.

This book presents an overview of the later medieval trinitarian theology of the rival Franciscan and Dominican intellectual traditions, and includes detailed studies of thinkers such as Thomas Aquinas, Henry of Ghent, John Duns Scotus, ...

This article offers the first critical edition of the most important version of Francis of Marchia's famous question 1 of his commentary on Book IV of the Sentences, in which the Franciscan theologian puts forth his virtus derelicta theory of projectile motion. The introduction attempts to place Marchia's theory in its proper context. The theory might seem to us an obvious improvement on Aristotle, but rather than an immediate and complete break with tradition that all scholastics quickly adopted, Marchia's virtus (...) derelicta was more a stage in a gradual process that had begun many decades before and did not find universal acceptance among his first successors. Moreover, Marchia himself did not take the theory to what might seem the obvious conclusion that Jean Buridan would draw, because Marchia employed the virtus derelicta to explain more phenomena than just projectile motion. (shrink)

Since 1991 the Franciscan Francis of Marchia, master of theology at the University of Paris (fl. 1320), has begun receiving his due attention as an exciting and innovative thinker. This volume examines his doctrines in cosmology, physics, metaphysics, ethics, and politics.

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

Using the Friedman-Koepke Hyperfine Structure Theory of [2], we provide a short construction of a gap 1 morass in the constructible universe.

The singular message in Global Health Law is that we must strive to achieve global health with justice—improved population health, with a fairer distribution of benefits of good health. Global health entails ensuring the conditions of good health—public health, universal health coverage, and the social determinants of health—while justice requires closing today’s vast domestic and global health inequities. These conditions for good health should be incorporated into public policy, supplemented by specific actions to overcome barriers to equity. A new global (...) health treaty grounded in the right to health and aimed at health equity—a Framework Convention on Global Health —stands out for its possibilities in helping to achieve global health with justice. This far-reaching legal instrument would establish minimum standards for universal health coverage and public health measures, with an accompanying national and international financing framework, require a constant focus on health equity, promote Health in All Policies and global governance for health, and advance the principles of good governance, including accountability. While achieving an FCGH is certainly ambitious, it is a struggle worth the efforts of us all. The treaty’s basis in the right to health, which has been agreed to by all governments, has powerful potential to form the foundation of global governance for health. From interpretations of UN treaty bodies to judgments of national courts, the right to health is now sufficiently articulated to serve this role, with the individual’s right to health best understood as a function of a social, political, and economic environment aimed at equity. However great the political challenge of securing state agreement to the FCGH, it is possible. States have joined other treaties with significant resource requirements and limitations on their sovereignty without significant reciprocal benefits from other states, while important state interests would benefit from the FCGH. And from integrating the FCGH into the existing human rights system to creative forms of compliance and enforcement and strengthened domestic legal and political accountability mechanisms, the treaty stands to improve right to health compliance. The potential for the FCGH to bring the right to health nearer universal reality calls for us to embark on the journey towards securing this global treaty. (shrink)

On August 30, 2013, the American Political Science Association sponsored a roundtable on political epistemology as part of its annual meetings. Co-chairing the roundtable were Jeffrey Friedman, Department of Government, University of Texas at Austin; and Hélène Landemore, Department of Political Science, Yale University. The other participants were Scott Althaus, Department of Political Science, University of Illinois at Urbana-Champaign; Mark Bevir, Department of Political Science, University of California at Berkeley; Rogers Smith, Department of Political Science, University of Pennsylvania; and (...) Susan Stokes, Department of Political Science, Yale University. We thank the participants for permission to republish their remarks, which they subsequently edited for clarity. (shrink)

Immanuel Kant's Metaphysical Foundations of Natural Science (1786) provides metaphysical foundations for the application of mathematics to empirically given nature. The application that Kant primarily has in mind is that achieved in Isaac Newton's Principia (1687). Thus, Kant's first chapter, the Phoronomy, concerns the mathematization of speed or velocity, and his fourth chapter, the Phenomenology, concerns the empirical application of the Newtonian notions of true or absolute space, time, and motion. This paper concentrates on Kant's second and third chapters—the Dynamics (...) and the Mechanics, respectively—and argues that they are best read as providing a transcendental explanation of the conditions for the possibility of applying the (mathematical) concept of quantity of matter to experience. Kant again has in mind the empirical measures of this quantity that Newton fashions in the Principia, and he aims to make clear, in particular, how Newton achieves a universal measure for all bodies whatsoever by projecting the static quantity of terrestrial weight into the heavens by means of the theory of universal gravitation. Kant is not attempting to prove a priori what Newton has established empirically but, rather, to clarify the character of Newton's mathematization by building Newton's empirical measures into the very concept of matter that is articulated in the Metaphysical Foundations. (shrink)

Peirce's Reality and Berkeley's Blunders LESLEY FRIEDMAN IN A NUMBER OF HIS LATE REMARKS, Peirce makes it clear that he holds Bishop Berkeley in the highest esteem. Hailed as the "father of all modern philoso- phy," Peirce argues that Berkeley, not Kant, "first produced an Erkenntnis- theorie, or 'principles of human knowledge', which was for the most part cor- rect in its positive assertions" ? This is not at all to say that Berkeley escapes rebuke; in spite of several (...) laudatory remarks, ~ Peirce takes him to task for two errors: his view of reality as actual perception, and his neglect of real possibility. Additionally, in some passages we find Berkeley both credited for some insight and admonished for some oversight. In one such occurrence, ' The following abbreviations are used throughout this paper: RFB "Review of Fraser's Berkeley," Nation 73 : 95-96. LSB Robert G. Meyers and Richard H. Popkin, eds., "Early Influences on Peirce: A Letter to Samuel Barnett,"Journal of the History of Philosophy 31 : 6o7-21. This letter is dated December 2o, 19o 9. CP Collected Papers of Charles Sanders Peirce, eds. C. Hartshorne and P. Weiss . and A. Burks . Cited by volume and paragraph number. MS Peirce's manuscripts from microfilm rolls a-3 o, following the Robin listing: Annotated Catalogue of the Papers of Charles S. Peirce (Amherst: University of Massachusetts Press,.. (shrink)

The appearance of consciousness in the universe remains one of the major mysteries unsolved by science or philosophy. Absent an agreed-upon definition of consciousness or even a convenient system to test theories of consciousness, a confusing heterogeneity of theories proliferate. In pursuit of clarifying this complicated discourse, we here interpret various frameworks for the scientific and philosophical study of consciousness through the lens of social insect evolutionary biology. To do so, we first discuss the notion of a forward test versus (...) a reverse test, analogous to the normal and revolutionary phases of the scientific process. Contemporary theories of consciousness are forward tests for consciousness, in that they strive to become a means to classify the level of consciousness of arbitrary states and systems. Yet no such theory of consciousness has earned sufficient confidence such that it might be actually used as a forward test in ambiguous settings. What is needed now is thus a legitimate reverse test for theories of consciousness, to provide internal and external calibration of different frameworks. A reverse test for consciousness would ideally look like a method for referencing theories of consciousness to a tractable model system. We introduce the Ant Colony Test as a rigorous reverse test for consciousness. We show that social insect colonies, though disaggregated collectives, fulfill many of the prerequisites for conscious awareness met by humans and honey bee workers. A long lineage of philosophically-neutral neurobehavioral, evolutionary, and ecological studies on social insect colonies can thus be redeployed for the study of consciousness in general. We suggest that the ACT can provide insight into the nature of consciousness, and highlight the ant colony as a model system for ethically performing clarifying experiments about consciousness. (shrink)

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and (...) that the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) (...) face complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

Erich Fromm was a political activist, psychologist, psychoanalyst, philosopher, and one of the most important intellectuals of the twentieth century. Known for his theories of personality and political insight, Fromm dissected the sadomasochistic appeal of brutal dictators while also eloquently championing love--which, he insisted, was nothing if it did not involve joyful contact with others and humanity at large. Admired all over the world, Fromm continues to inspire with his message of universal brotherhood and quest for lasting peace. The first (...) systematic study of Fromm's influences and achievements, this biography revisits the thinker's most important works, especially _Escape from Freedom_ and _The Art of Loving_, which conveyed important and complex ideas to millions of readers. The volume recounts Fromm's political activism as a founder and major funder of Amnesty International, the National Committee for a Sane Nuclear Policy, and other peace groups. Consulting rare archival materials across the globe, Lawrence J. Friedman reveals Fromm's support for anti-Stalinist democratic movements in Central and Eastern Europe and his efforts to revitalize American democracy. For the first time, readers learn about Fromm's direct contact with high officials in the American government on matters of war and peace while accessing a deeper understanding of his conceptual differences with Freud, his rapport with Neo-Freudians like Karen Horney and Harry Stack Sullivan, and his association with innovative artists, public intellectuals, and world leaders. Friedman elucidates Fromm's key intellectual contributions, especially his innovative concept of "social character," in which social institutions and practices shape the inner psyche, and he clarifies Fromm's conception of love as an acquired skill. Taking full stock of the thinker's historical and global accomplishments, Friedman portrays a man of immense authenticity and spirituality who made life in the twentieth century more humane than it might have been. (shrink)

I discuss three potential sources of evidence for truth in set theory, coming from set theory’s roles as a branch of mathematics and as a foundation for mathematics as well as from the intrinsic maximality feature of the set concept. I predict that new non first-order axioms will be discovered for which there is evidence of all three types, and that these axioms will have significant first-order consequences which will be regarded as true statements of set theory. The bulk of (...) the paper is concerned with the Hyperuniverse Programme, whose aim is to discover an optimal mathematical principle for expressing the maximality of the set-theoretic universe in height and width. (shrink)

Virtually every infant in the United States (U.S.) undergoes a heel stick within the first week of life to test for a variety of metabolic, endocrine, and hematological conditions as part of state-run universal newborn screening (NBS) programs. In the U.S., NBS began in the 1960s for phenylketonuria (PKU), a metabolic condition that causes intellectual disability if left untreated. I review the history of how NBS came to be a mandatory public health program that did not require parental consent1 and (...) examine whether the policy was morally justifiable. I then examine how three changes to NBS programs are prompting a re-evaluation of the mandatory nature of NBS. The three changes are: (1) the implementation of .. (shrink)

In August 2000, Arthur Matas and his colleagues de scribed a protocol in which their institution began to accept as potential donors, individuals who came to the University of Minnesota hospital offering to donate a kidney to any patient on the waiting list. Matas and his colleagues refer to these donors as nondirected donors by which is meant that the donors are altruistic and that they give their organs to an unspecified pool of recipients with whom they have no emotional (...) relationship. This paper represents an ethical and policy critique of the nondirected donation protocol that was implemented at the University of Minnesota in August 1999. Specifically, I address the ethical questions: Whether altruistic living solid organ donations by strangers should be permitted? And if so, What are appropriate ethical guidelines for such donations? (shrink)

by Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ April 5, 1997..

. In this paper, I attempt to dissolve the theism/atheism boundary. In the first part, I consider last things, according to mainstream science. In the second part, I define the Natural God as the Force of Nature—evolving, unifying, maximizing—and consider Its relation to last things. Finally, I discuss our knowledge of the Natural God and Its relevance to our personal lives. I argue that we can know the Natural God through scientific reason combined with global intuition, and that this knowledge, (...) from the perspective of last things, may help us achieve universal love, ethical action, and personal salvation. (shrink)

We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a (...) proper class of supercompact cardinals. Finally, we also produce a model in which the unique supercompact cardinal is also the only strongly compact cardinal, no cardinal is supercompact up to an inaccessible cardinal, level by level inequivalence holds and the unique supercompact cardinal is not HOD-supercompact. (shrink)

Vopenka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over, and indeed over the even smaller inner model $\mathbb{S}=$, where S is the Stability predicate. We refer to the inner model $\mathbb{S}$ as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible (...) to add reals which are not set-generic but preserve the Stable Core. (shrink)