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)

Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$ , there is a supertransitive inner model of Zermelo containing all ordinals in which for every λ A (...) λ = {α ∣Φ(λ, a)}. (shrink)

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and Shoenfield’s unramified (...) forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being that in the title, not counting 1 179 618 517 981 links between symbols that are needed to disambiguate the whole expression.

Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being that in the title, not counting 1 179 618 517 981 links between symbols that are needed to disambiguate the whole expression.

Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula $\Phi$ such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$, there is a supertransitive inner model of Zermelo containing all ordinals in which for every $\lambda A_{\lambda} = \{\alpha (...) \mid\Phi\}$. (shrink)

We derive the schemes of from certain weak forms of the same.

We prove the existence of p-ideals that are nonmeagre subsets of P(ω) under various set-theoretic assumptions.

1. The “puzzle” Physical objects are coloured: roses are red, violets are blue, and so forth. In particular, physical objects have fine-grained shades of colour: a certain chip, we can suppose, is true blue (unique, or pure blue). The following sort of scenario is commonplace. The chip looks true blue to John; in the same (ordinary) viewing conditions it looks (slightly) greenish-blue to Jane. Both John and Jane are “normal” perceivers. Now, nothing can be both true blue and greenish-blue; since (...) the chip is true blue, it is not greenish-blue. Hence Jane, unlike John, is misperceiving the chip. Generalizing, the conclusion is that there is widespread misperception of fine-grained shades. According to Tye (2006), and Cohen, Hardin, and McLaughlin (2006), the previous paragraph amounts to a paradox: an apparently unacceptable conclusion has been drawn from apparently acceptable premises via apparently acceptable reasoning. (See also Hawthorne and Kovakovich 2006, 180-1.) Tye swallows the conclusion, aided by a dose of evolutionary speculation. Hardin (1988), on the other hand, rejects the first premise, and denies that physical objects are coloured. Cohen (2004) and McLaughlin (2003) claim that both Jane and John have the colour of the chip right. Our opening paragraph concealed a crucial parameter. In fact, the chip looks greenish-blue-relative-to- circumstances-C to Jane, and true-blue-relative-to-circumstances-C* to John, and the chip has both these relativized colours.1 All this ingenious philosophizing would be in vain, of course, if the conclusion of the opening paragraph were not puzzling or problematic. So, why is it supposed to be? According to Tye, the conclusion is puzzling because John and Jane are both “normal perceivers” (xx). He seems to think that it is (prima facie) plausible to assume.. (shrink)

Initial responses to questionnaires used to assess participants' understanding of informed consent for malaria vaccine trials conducted in the United States and Mali were tallied. Total scores were analyzed by age, sex, literacy (if known), and location. Ninety-two percent (92%) of answers by United States participants and 85% of answers by Malian participants were correct. Questions more likely to be answered incorrectly in Mali related to risk, and to the type of vaccine. For adult participants, independent predictors of higher scores (...) were younger age and female sex in the United States, and male sex in Mali. Scores in the United States were higher than in Mali (P = 0.005). Despite this difference participants at both sites were well informed overall. Although interpretation must be qualified because questionnaires were not intended as research tools and were not standardized among sites, these results do not support concerns about systematic low understanding among research participants in developing versus developed countries. (shrink)

The export of processed RNA molecules from the nucleus is an intricately regulated process, subject to various developmental and physiological controls. The structural and biochemical properties of the nuclear envelope that are involved in this process are described here.

B. R. Frieden uses a single procedure, called extreme physical information, with the aim of deriving 'most known physics, from statistical mechanics and thermodynamics to quantum mechanics, the Einstein field equations and quantum gravity'. His method, which is based on Fisher information, is given a detailed exposition in this book, and we attempt to assess the extent to which he succeeds in his task.

Copyright © 2015 by the Society of Criti. Objective: Global lung stress varies considerably with low tidal volume ventilation for acute respiratory distress syndrome. High stress despite low tidal volumes may worsen lung injury and increase risk of death. No widely available parameter exists to assess global lung stress. We aimed to determine whether the volume delivered during a recruitment maneuver is inversely associated with lung stress and mortality in acute respiratory distress syndrome. Design: Substudy of an acute respiratory distress (...) syndrome clinical trial on esophageal pressure-guided positive end-expiratory pressure titration. Setting: U.S. academic medical center. Patients: Forty-two patients with acute respiratory distress syndrome in whom airflow, airway pressure, and esophageal pressure were recorded during the recruitment maneuver. Interventions: A single recruitment maneuver was performed before initiating protocol-directed ventilator management. Recruitment maneuvers consisted of a 30-second breath hold at 40 cm H2O airway pressure under heavy sedation or paralysis. VRM was calculated by integrating the flow-time waveform during the maneuver. End-inspiratory stress was defined as the transpulmonary pressure during end-inspiratory pause of a tidal breath and tidal stress as the transpulmonary pressure difference between end-inspiratory and end-expiratory pauses. Measurements and Main Results: VRM ranged between 7.4 and 34.7 mL/kg predicted body weight. Lower VRM predicted high end-inspiratory and tidal lung stress. After adjusting for Pao2/Fio2 and either driving pressure, tidal volume, or plateau pressure and positive end-expiratory pressure, VRM remained independently associated with both end-inspiratory and tidal stress. In unadjusted analysis, low VRM predicted increased risk of death. VRM remained significantly associated with mortality after adjusting for study arm. Conclusions: Low VRM independently predicts high lung stress and may predict risk of death in patients with acute respiratory distress syndrome. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 12.4 (2005) 301-305 [Access article in PDF] The Beauty of Psychotherapy R. D. Hinshelwood Keywords awe, psychotherapy, representation, self-esteem The Enlightenment was devoted to clear uncontaminated reason; its success has given us the terrific achievements of science and technology. However, it has bequeathed problems too. Untrammeled reason has led to the devaluing and exclusion of emotions. Emotions are irrational—self-deception, akrasia, and so on. They were (...) a problem for the eighteenth-century philosopher Kant and right through to the modern therapist Aaron Beck. Callender rightly looks around the edges of rationalist thought to see what can be said about the emotional and subjective side of mankind. He argues that there is more to the human mind than cognition. Something has been left out. What can be found to fill the gap?Callender has searched for the missing link that might complete our view of human nature. And he came back with aesthetics. Meaning The Enlightenment failing is writ large for therapists who are confronted with people who feel things. So we need to ask what place the emotions have in understanding the nature of man? It is not a hypothetical question, but a very practical one for a therapist. Emotions, Callender says, and most people would agree with him, are what gives meaning to life, a sense of purpose in living, even an ill-defined or delusional purpose.He presents Kant's description of aesthetics. Aesthetic judgements have a number of features, the most enigmatic and perhaps profound is a "purposiveness without purpose." A work of art, say a portrait, may have a purpose insofar as it represents a person for his aggrandizement or for his descendents through many generations, but the aesthetic quality does not come from that purpose. The aesthetic purpose involves a different set of judgements. They are about the painting as a piece of art in itself. If it is possible to compare this kind of aesthetic judgement with a judgement of a person's life, then I find myself reminded of Wollheim's definition of a person: The simplest way of putting the matter, which is not to be taken as a metaphysical claim, is this: There are persons, they exist; persons lead lives; and as a result, in consequence—in consequence that is, of the way they do it—there are lives, of which those who lead them may, for instance, be proud, or feel ashamed. So there is a thing, and there is a process, and there is a product. (Wollheim 1984, 2) Although many things do not lead lives, we do. And our lives have a process and a product. Like an aesthetic product, our lives have a sense of emotional meaning, without it having a clear purpose.Aesthetic judgements are therefore of a particular and somewhat peculiar kind. They judges talent for producing "that for which no determinate [End Page 301] rules can be given," as the paper quotes Kant. Other kinds of judgements do have determinate rules. First are empirical judgements for getting practical things done; and moral judgements for assessing ethical good from bad. Aesthetic beauty does not have clear rules, although many systems of rules have been proposed starting with the rules of perspective as they were laid out by renaissance painters, through to systems of color, such as that proposed by Paul Klee, once color was correlated with electromagnetic wavelengths. However, artists have always kept one jump ahead and evaded imprisonment in codified rules. So too it is with leading a life, perhaps. The really creative life is not lead according to rules. The art of life is to live meaningfully without simply running along tramlines. Although moral conduct and the practicalities of life remain significant, perhaps leading a life implies simply a "flare for living." Representation Of these three kinds of judgements that a mind makes, two do have sets of rules. Only one based on rationality, empirical evidence, and reason. Reason judges what can be done and achieved in the real physical world. It is about pragmatics and the usefulness of action. Moral judgements, about what... (shrink)