We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong , hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we (...) refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author’s outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals. (shrink)

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.

Over the past forty years, the health humanities, previously called the medical humanities, has emerged as one of the most exciting fields for interdisciplinary scholarship, advancing humanistic inquiry into bioethics, human rights, health care, and the uses of technology. It has also helped inspire medical practitioners to engage in deeper reflection about the human elements of their practice. In _Health Humanities Reader_, editors Therese Jones, Delese Wear, and Lester D. Friedman have assembled fifty-four leading scholars, educators, artists, and clinicians (...) to survey the rich body of work that has already emerged from the field—and to imagine fresh approaches to the health humanities in these original essays. The collection’s contributors reflect the extraordinary diversity of the field, including scholars from the disciplines of disability studies, history, literature, nursing, religion, narrative medicine, philosophy, bioethics, medicine, and the social sciences. With warmth and humor, critical acumen and ethical insight, _Health Humanities Reader_ truly humanizes the field of medicine. Its accessible language and broad scope offers something for everyone from the experienced medical professional to a reader interested in health and illness. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

This article describes the evolution of extra-mural team projects in the Wharton School's new MBA curriculum, emphasizing both the benefits of doing community service and the value these projects have in providing real work opportunities for learning teams; five- or six-person student groups that do collective tasks in Foundations of Leadership and other required courses throughout the first of a two-year program.

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.

We lift Jensen's coding method into the context of Woodin cardinals. By a theorem of Woodin, any real which preserves a "strong witness" to Woodinness is set-generic. We show however that there are class-generic reals which are not set-generic but preserve Woodinness, using "weak witnesses".

Jan Narveson criticizes the view expressed in my Libertarian Philosophy in the Real World that there is no orthodox libertarian position on the ethics of abortion. He asserts that fetuses lack the defining characteristics of personhood, and thus are ineligible for what he terms “intrinsic” rights under his, and presumably any other, plausible libertarian theory. My counterargument is threefold: Narveson’s contractarianism can be interpreted in a way that is consistent with the pro-life perspective; because his theory permits no principled distinction (...) between the moral status of third trimester fetuses and newborns, the contrary reading of his social contract produces a result that is implausible and even repellent; and even if his version of contractarianism does imply a unique, aggressively pro-choice stance on abortion, there are competing libertarian theories that are receptive to pro-life views. (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.

The book attempts to develop an account of simplicity in terms of testability, and to use this account to provide an adequate characterization of induction, one immune to the class of problems suggested by Nelson Goodman. It is then shown that the past success of induction, thus characterized, constitutes evidence for its future success. A qualitative measure of confirmation is developed, and this measure - along with the considerations of simplicity - is used to provide an account of the consilience (...) of inductions, and also an inductivist account of the structure and progress of scientific theory. An appendix extends the treatment of simplicity to statistical distributions and provides a reasonable interpretation of the maximum entropy principle. Thus, this book is an attempt to characterize induction in terms of a well-defined notion of simplicity and to use that characterization as a basis of an account of empirical, and in particular, scientific reasoning. (shrink)

E. T. Jaynes' resolution of Bertrand's paradox in terms of invariance principles is criticized. An experimental setup is considered which generates general solutions to Bertrand's problem by rotating a line around a point a distancer+d from a circle of radiusr. The general solution obtained is neither translationally nor scale invariant, but depends on the value ofr/d. Only in the limitr/d » 0, when the line is just translating across the circle, is the distribution translationally invariant and scale invariant. In this (...) limiting case the distribution found is just that obtained by Jaynes. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

This essay argues that film deserves a place within the medical humanities curriculum and demonstrates effective strategies for employing it within medical ethics and humanities classrooms. Part One of the article emphasizes how and why medical ethics teachers can utilize documentary and fictional films, such as “Thomas Szasz and the Myth of Mental Illness,” “The Deadly Deception,”Whose Life Is It Anyway? and “Voices From the Front” in their courses. Such films encourage students to move beyond abstract debates and confront the (...) human pain inherent in all ethical dilemmas. Part Two focuses on documentary and fiction film in the medical humanities classroom. In this section, the author details how to incorporates films, such asThe Doctor, The Waterdance andHospital, into the humanities classroom, juxtaposing them with various literary works, such asOther Women's Children, Borrowed Time, andCeremony. Part Three of the essay presents a detailed discussion ofThe Elephant Man andFrankenstein, illustrating how visual and literary texts compliment each other within the humanities classroom. Overall, the author demonstrates how films function as engaging and complex visual texts providing unique insights in the particularities of American health care and, as such, can become valuable components within medical ethics and humanities classrooms. (shrink)

Let ℜ be an o-minimal expansion of (ℝ, <+) and (φk)k∈ℕ be a sequence of positive real numbers such that limk→+∞f(φk)/φk+1=0 for every f:ℝ→ ℝ definable in ℜ. (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of φ.

In lieu of an abstract, here is a brief excerpt of the content:Reply to Flage Lesley Friedman "The chiefexercise of the memory," Hume tells us, "is not to preserve the simple ideas, but their order and position."1 On Daniel Flage's interpretation ofHume, however, it is the only exercise ofthe memory (1985a, 1985b, 1990). Flage's account can accommodate onlymemories of complex ideas; he disallows the possibility of 'preserving* a simple idea in its simplicity. Yet there is an example of a (...) purely simple impression in the Treatise; and since for every impression there is in principle a corresponding idea which we can later recall, this example establishes the existence of a purely simple memory-idea. In what follows I will provide support for this claim, Flage's counterarguments (this volume) notwithstanding. Flage argues that given any perceived object, say a coloured dot, there is a visual field in which we perceive it. This field and the impression we receive from it, "must be a complex impression, that is, an impression that is composed of indefinitely many colour points. In this respect it is analogous to a pointillist painting or, to use a more contemporary example, a computer array composed of pixels" (1993, 190). Flage is probably right about this; from our point of view, it is implausible to suggest that there is any purely simple background, or worse, that there is no impression ofa background at all. Nonetheless, my concern is with what Hume thought and not with what is plausible orimplausible; and Hume does give an example ofa "background" that is notcomplex, namely, the scenarioinwhichwe perceive points oflight in darkness. We may observe, that when two bodies present themselves, where there wasformerly an entire darkness, the onlychange, that is discoverable, is in the appearance ofthese two objects, and that all the rest continues to be as before, a perfect negation oflight, and of every colour'd or visible object. This is not only true of what may be said to be remote from these bodies, but also of the very distance; which is interpos'd betwixt them; that being nothing but darkness, or the negation of light; without parts, without composition, invariable and indivisible. (T 57, bold italics added) I originally considered the ink spot experiment and theidea ofred in the dark as examples ofpurely simple impressions. Let me try this Volume XLX Number 1 199 LESLEYFRIEDMAN again, elaborating a bit more on the latter (what I take to be Hume's example of a version of the ink spot experiment). Early on in the Treatise Hume speaks of"that idea ofred which we form in the dark" (T 3). Let us then consider a red spot in complete darkness. Flage and I agree that this spot is qualitatively simple. Our disagreement is rooted in the nature ofthe "background," in this case the darkness. Flage claims that "the sighted-person sitting in the dark has an impression or impressions, namely, an impression of darkness" (1993, 190); and therefore the impression we have of the spot of red in darkness is not purely simple. In support of this claim he cites the followingevidence: Hume's remarks that (1) "the sighted-person sitting in the dark "receives no other perception... than what is common to him with one born blind' " (ibid., 189; T 55-56), and (2) "the idea of utter darkness can never be the same with that of vacuum" (T 56). Let me take each ofthese in turn. Presumably, the first quote is cited in order to estabUsh that the blind person does indeed have a perception (impression) of darkness. This cannot be right; ifsuch a person had an impression ofdarkness, he would also have an idea of darkness, and if we look to the part of the sentence which Flage leaves out of his quotation, we find the following: "and 'tis certain such-a-one [that is, a blind person] has no idea either of light or darkness" (T 56). Hume's point is simple: the reason that the blind person has no idea ofdarkness is because he has no impression of darkness; and we are in the same position as him when we are sitting in the dark. Nor does the second quote provide evidence for the... (shrink)

This CQ department is dedicated to bringing noted bioethicsts together in order to debate some of the most perplexing contemporary bioethics issues. You are encouraged to contact department editor, D. Micah Hester, UAMS/Humanities, 4301 W. Markham St. #646, Little Rock, AR 72205, with any suggestions for debate topics and interlocutors you would like to see published herein.

The first principle of the Nuremberg Code requires the informed consent of the subject. Proxy consent was not addressed until the Declaration of Helsinki. U.S. policies regarding consent for the participation of children in research would not be finalized for almost two more decades in subpart D of the federal regulations that govern the participation of subjects in research. In October 2000, the Children's Health Act was passed. Title X required the Secretary of the Department of Health and Human Services (...) to conduct a review of the regulations under subpart D and “to consider any modifications necessary to ensure the adequate and appropriate protection of children participating in research, and report the findings of the Secretary to Congress.” In this article, I provide a brief overview of the current informed consent requirements in pediatric research, which include requirements for parental permission and the child's assent. I then examine the moral bases for these requirements; whether the current policies are consistent with these moral foundations; and if not, how the policies ought to be modified. a. (shrink)

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the (...) generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d (κ) is less than 2κ for inaccessible κ, where d (κ) is the dominating number at κ) is internally consistent, given the existence of 0#. (shrink)

This article presents a critical edition from the six surviving witnesses of Landulph Caracciolo’s , Scriptum in I Sententiarum, d. 23, a text that has never appeared in print before. A short introduction begins to set Landulph’s treatment of intentions and intentionality in this text into its historical, philosophical, and theological context, in particular linking it to the positions of John Duns Scotus and Peter Auriol.

In lieu of an abstract, here is a brief excerpt of the content:Another Look at Flage's Hume Lesley Friedman In recent articles, Daniel Flage (1985a, 1985b) offers an interpretation of Humean memory-ideas as relative ideas: ideas of memory are analogous to definite descriptionsinsofar as they single outexactly one entity.1 Consequently, Flage argues that Hume has provided an adequate distinction between ideas generated by memory and ideas generated by imagination. It is my contention that Flage's reading is neither consonant with (...) Hume's remarks in the Treatise nor successful in reducing the number ofdifficulties with Hume's theory ofmemory. I shall argue that (1) it is not clear that all memory-ideas are complex (an assumption necessary for Flage's interpretation); (2) Flage construes all memory as episodic, but he overlooks semantic memory; and (3) there isa distinction between mis-remembering and imagining which Flage's account cannot accommodate. It is worth noting that Flage contends that he is giving an interpretation ofHume's account ofmemory; he is not simply attempting to find out the truth ofthe matter. That is, Flage's position is that there is good reason to think that Hume construed memory-ideas as relative ideas. Accordingly, Flage must provide evidence, preferably textual, for the claim that Hume held such a view. My position is thatit is unlikely that Hume held such a view, and even if he did, it is inadequate for grounding an external distinction between memory-ideas and imagination-ideas. In short, claiming that Hume was interested in a decisive distinction between ideas ofmemory and ideas ofimagination is to involve Hume in problems with which he was not concerned. Not only are Flage's remarks in conflict with Hume's, but ifwe readhim as Flage suggests, Hume's theory ofmemory becomes considerably more riddled with problems than it is if we do not so read him. Flage and Hume In section 1 of Treatise 1.1, Hume draws his celebrated distinction between ideas and impressions. In the second section he considers impressions in some detail and he does the same for ideas in section three. Following every impression, Hume claims, there is a corresponding idea, either produced by the faculty of memory or the faculty of imagination. He takes it as no less than obvious that memory-ideas are more "lively and strong" than imagination-ideas: Volume XDi Number 1 177 LESLEY FRIEDMAN Tis evident at first sight, that the ideas of the memory are much more lively and strong than those of the imagination, and that the former faculty paints its objects in more distinct colours, than any which are emplo/d by the latter.2 That memory-ideas are more lively than their counterparts in imagination Hume construes as asensible difference between the two. Accordingly, I shall refer to this difference as the internal difference. Thereis,however,another difference betweenthesekinds ofideas, namely, that memory-ideas preserve the order and form of their corresponding simple ideas, while imagination-ideas need not: Tis evident, that the memory preserves the original form, in which its objects were presented, and that where-ever we depart from it in recollecting any thing, it proceeds from some defect or imperfection in that faculty.... The chiefexercise of the memoryis not to preserve the simpleideas, but their order and position.... The same evidence follows us in our second principle, of the Uberty ofthe imagination to transpose andchange its ideas. (T 9-10) I shall refer to the above as the external difference between memory-ideas andimagination-ideas.3 During his discussion ofcausation, Hume elaborates on the above differences. The external difference, he claims, "is not sufficient to... make us know the one from the other; it being impossible to recai the pastimpressions, in order to compare them with our present ideas, and see whether theirarrangementbe exactlysimilar" (T85). He concludes that the difference between memory and imagination is simply that memory-ideas are more forceful and vivacious than imagination-ideas. In short, since the external difference is non-phenomenal, it is inadequate for distinguishing, in practice, between ideas that are the product ofmemory and ideas that are the product of the imagination. Before concluding his discussion of the above differences Hume voices anote ofscepticismconcerningthe internal difference. He argues that theinternal... (shrink)

Aristotle's doctrine of the four causes--formal, final, efficient and material--is famous. But Posterior Analytics B 11 lists "if certain things hold, it is necessary that this does" in place of a standard expression for the material cause. This cause has been dubbed the grounding cause. It has interested scholars since the Greek commentators, who simply assumed that Aristotle meant the material cause. This traditional thesis has been challenged by two views: first, that the grounding cause is a special type of (...) material cause ; second, that it is no material cause at all . ;I set out two arguments for the traditional thesis and consider nine objections to it. Since most of these objections, which constitute the support for the revisionist thesis, can be answered, I reject the revisionist thesis. But in my treatment of one objection it also becomes clear that the traditional thesis is mistaken. The grounding cause is a necessitating cause, yet in Physics B 9 Aristotle makes it clear that many material causes are not necessitating causes. Thus, some form of the modified traditional thesis is true. ;But which form? D. M. Balme suggests that Aristotle means to include only mathematical causes because hypothetical necessity covers all of the relation between matter and product in things which come to be. But Aristotle seems to accept necessitating material causes in several passages in the biological works. In other passages he seems to adopt the position Balme attributes to him. I argue that he does not actually adopt this position, and that he does accept simple necessity in some things which come to be. Thus, the grounding cause is a necessitating material cause, and such causes are found not only in mathematics, but also among things which come to be. (shrink)

Nella teologia trinitaria di Enrico di Gand, e in particolare nell'uso della nozione di verbum mentis, confluiscono ad avviso dell'A. tre diverse tradizioni che vengono rielaborate in un sistema coerente dal maestro agostiniano. La prima, quella cui aderiscono Tommaso e Bonaventura, insiste sul fatto che le persone sono distinte tra loro proprie per mezzo di relazioni opposte. La seconda, rappresentata da Riccardo di san Vittore, che attribuisce la differenza tra le persone al loro diverso modo di emanazione e di origine. (...) La terza che riprende il modello psicologico della trinità di Agostino ed in particolare l'immagine del Figlio come Verbum. Enrico incorpora i diversi elementi di queste tradizioni in modo originale e sostiene che il Figlio è distinto dalle altre due persone in base al diverso modo di emanazione , ma che è diverso dal Padre e dallo Spirito Santo anche perché è Verbum. L'A. sottolinea la notevole influenza esercitata da Bonaventura sul pensiero trinitario di Enrico, anche dal punto di vista della terminologia utilizzata e ipotizza che ciò sia dovuto al ruolo di «propaganda» delle dottrine bonaventuriane svolto da Ruggero Marston e Matteo d'Aquasparta. (shrink)

S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel, A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman, Solecki, R. Spatzier, J. Steel, D. Sullivan, S. (...) Thomas, A. Valette, V.S. Varadarajan, B. Velickovic, B. Weiss, J.D.M. Wright, R.J. Zimmer. (shrink)

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)

Several physicists, among them Hawking, Page, Coule, and Carroll, have argued against the probabilistic intuitions underlying fine-tuning arguments in cosmology and instead propose that the canonical measure on the phase space of Friedman-Robertson-Walker space-times should be used to evaluate fine-tuning. They claim that flat space-times in this set are actually typical on this natural measure and that therefore the flatness problem is illusory. I argue that they misinterpret typicality in this phase space and, moreover, that no conclusion can be (...) drawn at all about the flatness problem by using the canonical measure alone. (shrink)

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...) adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (shrink)

Through most of the history of economics, the most influential commentators on methodology were also eminent practitioners of economics. And even not so long ago, it was so. Milton Friedman, Paul Samuelson, Trygve Haavelmo, and Tjalling Koopmans were awarded Nobel prizes for their substantive contributions to economics, and were each important contributors to methodological thought. But the fashion has changed. Specialization has increased. Not only has methodology become its own field, but many practitioners have come to agree with Frank (...) Hahn's (1992) view that methodology is a distraction to the practitioner, best left to the professional methodologists and philosophers, and of little practical import even when delivered from their pens. John Sutton's lectures, Marshall's Tendencies: What Economists Can Know, is a welcome return to the older fashion, for Sutton is an eminent practitioner of game theory and industrial organization. One of the main themes of these rich and nuanced lectures is the relationship of economic theory to econometric evidence. Sutton's reflections on econometrics appear to arise from the darker recesses of his practitioner's soul. While he affects a sunny disposition and ends on a hopeful note, his analysis articulates the lurking fear that econometrics is a hopeless project and that economics has little to learn from the interaction of theory and econometrics. Sutton's book is like a play in which virtue triumphs, but the villain gets all the good lines. (shrink)

By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left open by (...) Sieg [23] and Hájek [9]. We also give a new proof of independence of $B\Sigma_{n+1}$ over $I\Sigma_n$ by a direct recursion-theoretic argument and answer an open problem formulated by Gaifman and Dimitracopoulos [8]. (shrink)