We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...) Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”. (shrink)

In this paper, we show that certain choiceless models of ZF originally constructed using an almost huge cardinal can be constructed using cardinals strictly weaker in consistency strength.

Using Steel's recent result that assuming AD, in L[R] below Θ, κ is regular $\operatorname{iff} \kappa$ is measurable, we mimic below Θ certain earlier results of Gitik. In particular, we construct via forcing a model in which all uncountable cardinals below Θ are singular and a model in which the only regular uncountable cardinal below Θ is ℵ 1.

If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded (...) in κ. The two aforementioned phenomena, however, need not occur in a universe with an indestructibly supercompact cardinal and sufficiently few large cardinals. In particular, we show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry fewer than the maximal number of normal measures. We also, however, show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry the maximal number of normal measures. If we weaken the requirements on indestructibility, then this last result can be improved to obtain a model with an indestructibly supercompact cardinal κ in which every measurable cardinal δ < κ carries the maximal number of normal measures. (shrink)

Action explanations like ‘I am heading to the ferry because the bridge is closed,’ are supposed to require restatement: ‘I am... because I believe the bridge is closed,’ because (i) the objective claim may be false though the intended explanation is correct, and (ii) because objective circumstances have to be cognitively mediated if they are to bear on action. This supposition is rejected here. Restatements cannot withdraw the objective claim without withdrawing the explanation. In the context of reason‐giving, belief statements (...) do not function as assertions about the inner life. The fact that belief statements may be true though the belief itself is false creates the illusion of distinctive subject matters. The illusion is exploited by conceptions of belief positing stored representations. Even if there were such representations, retrieved for the sake of expressing previously acquired beliefs, the expression of belief must endorse p and cannot merely advert to the existence and content of a representation. The reverse of the standard philosophical view about restatement is actually correct. Wherever an agent correctly adduces a belief that an objective circumstance obtains in explaining his action, a de‐psychologizing restatement that merely makes the objective claim must be ascribable to the agent. (shrink)

Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ κ is inaccessible. We show it is also (...) the case that if κ is indestructibly supercompact, then Vκ⊨T1, so by reflection, B1 = df{δ κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, Vδ⊨T1. It is thus not possible to prove in ZFC that Bi = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and Vδ⊨Ti} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide.

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P V so that for , There is a proper class of compact cardinals + If (...) ƒ = 0, then the αth compact cardinal is not supercompact + If ƒ = 1, then the αth compact cardinal is supercompact”. We then prove a generalized version of this theorem assuming κ is a supercompact limit of supercompact cardinals and ƒ : κ → 2 is a function, and we derive as corollaries of the generalized version of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of nonsupercompact strongly compact cardinals and the consistency of the least supercompact cardinal being a limit of strongly compact cardinals. (shrink)

This essay discusses how two physicians in Britain's National Health Service describe and analyze the conditions of their work: how algorithms and protocols structure the care they can provide and create the dilemmas they and their patients face. In these issues, the NHS is a canary in the mineshaft of contemporary Western health care. NHS practices are understood as how states and state-like entities, Leviathans, seek to render their subjects _legible_; in this instance, to make both physicians and patients transparently (...) visible to surveillance and administration by standardizing medical work and patient need. Physicians respond by engaging in workarounds, finding ways to provide care despite systemic restrictions. (shrink)

During Abraham Nussbaum's first year of medical school, he participated in a white coat ceremony and was invested, literally, with a white coat that is symbolic of entry into the medical profession. He was also given a book, an anthology of writings on medicine that Nussbaum describes as having a "wistful quality" and being "engaging but reverential" ; the dust jacket featured a Norman Rockwell painting. He later went to a second-hand bookstore and traded the anthology for Abraham Verghese's 1994 (...) memoir, My Own Country, an excerpt from which was in the anthology. For Nussbaum, struggling to figure out what it means to be a doctor, Verghese's writing "portrays the physician as the storyteller of his... (shrink)

Increasingly, medicine is used to remodel, revise, and revamp as much as to heal and mend. It is tempting to say that people make merely personal choices about these new uses. But such choices have implications for everybody, and they ought to be made cautiously, slowly, and in a way that opens them to discussion.

The narrative ethicist imagines life as multiple points of view, each reflecting a distinct imagination and each more or less capable of comprehending other points of view and how they imagine. Each point of view is constantly being acted out and then modified in response to how others respond. People generally have good intentions, but they get stuck realizing those intentions. Stories stall when dialogue breaks down. People stop hearing others' stories, maybe because those others have quit telling their stories. (...) The narrative ethicist's job is to help people generate new imaginations that can restart dialogues. (shrink)

We construct two models in which the least strongly compact cardinal κ is also the least strong cardinal. In each of these models, κ satisfies indestructibility properties for both its strong compactness and strongness.

We prove a consistency result about square principles and stationary reflection which generalises the result of Ben-David and Magidor [4].

We show that it is consistent, relative to a supercompact limit of supercompact cardinals, for the least strongly compact cardinal k to be both the least measurable cardinal and to be > 2k supercompact.

We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals.

Contemporary health care often lacks generosity of spirit, even when treatment is most efficient. Too many patients are left unhappy with how they are treated, and too many medical professionals feel estranged from the calling that drew them to medicine. Arthur W. Frank tells the stories of ill people, doctors, and nurses who are restoring generosity to medicine--generosity toward others and to themselves. The Renewal of Generosity evokes medicine as the face-to-face encounter that comes before and after diagnostics, pharmaceuticals, (...) and surgeries. Frank calls upon the Roman emperor Marcus Aurelius, philosopher Emmanuel Levinas, and literary critic Mikhail Bakhtin to reflect on stories of ill people, doctors, and nurses who transform demoralized medicine into caring relationships. He presents their stories as a source of consolation for both ill and professional alike and as an impetus to changing medical systems. Frank shows how generosity is being renewed through dialogue that is more than the exchange of information. Dialogue is an ethic and an ideal for people on both sides of the medical encounter who want to offer more to those they meet and who want their own lives enriched in the process. The Renewal of Generosity views illness and medical work with grace and compassion, making an invaluable contribution to expanding our vision of suffering and healing. (shrink)

Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as well. The large cardinal (...) hypotheses on λ are necessary, as we further demonstrate by constructing via forcing two distinct models in which either equation image or equation image. In each of these models, there is an indestructibly supercompact cardinal κ, and a restricted large cardinal structure above κ. If we weaken the indestructibility requirement on κ to indestructibility under partial orderings which are both κ-directed closed and -distributive, then it is possible to construct a model containing a supercompact cardinal κ witnessing this degree of indestructibility in which every measurable cardinal δ < κ is δ+ supercompact. (shrink)

It is known that if $\kappa < \lambda$ are such that κ is indestructibly supercompact and λ is 2λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible (...) under κ-directed closed forcing. (shrink)

1. Introduction. It is generally admitted that a large part of man's knowledge is based on inductive arguments. Hence any philosophical theory concerning the nature of inductive arguments constitutes an epistemological theory. Any such philosophical theory of induction must, if it is to be satisfactory, take adequate account of Hume's criticism of inductive arguments. One way of treating his criticism is to say that the validity of inductive arguments is in an important sense relative to some broad factual assumptions about (...) the general nature of the universe and that these general factual assumptions are presupposed in a certain way by the users of inductive arguments. Let us call any theory of this general type a postulate theory of induction. (shrink)

Hermann Lotze has truly said that “among all the errors of the human mind” the “strangest” is doubting its “own existence,” or regarding it “at second hand as the product of an external Nature” which can be known only “indirectly” while the mind knows itself directly. Yet this denial is found both in the Occident and in the Orient. Moreover, while in the latter it stems largely from an extreme form of idealism in terms of a reductionistic pantheism, in the (...) former it is almost altogether the result of an analytical, reductive naturalism. More specifically, in the West this impersonalistic tendency has really been the result of two movements which have merged in certain contemporary schools. (shrink)

We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary.

Assuming Con, a model in which there are unboundedly many regular cardinals below Θ and in which the only regular cardinals below Θ are limit cardinals was previously constructed. Using a large cardinal hypothesis far beyond Con, we construct in this paper a model in which there is a proper class of regular cardinals and in which the only regular cardinals in the universe are limit cardinals.

Starting from cardinals κ κ is measurable, we construct a model for the theory “ZF + n < ω[DCn] + ω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ω + 1, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.

We prove three theorems which show that it is relatively consistent for any strong cardinal κ to be fully Laver indestructible under κ-directed closed forcing.

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ + strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and (...) is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ +, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ + strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ +, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ +, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ +, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ + strongly compact. (shrink)

We establish two theorems concerning strongly compact cardinals and universal indestructibility for degrees of supercompactness. In the first theorem, we show that universal indestructibility for degrees of supercompactness in the presence of a strongly compact cardinal is consistent with the existence of a proper class of measurable cardinals. In the second theorem, we show that universal indestructibility for degrees of supercompactness is consistent in the presence of two non-supercompact strongly compact cardinals, each of which exhibits a significant amount of indestructibility (...) for its strong compactness. (shrink)