This narrative provides insight into medical education for Black physicians in South Carolina in the 1960s, during the civil rights movement. It also discusses the many rewards and challenges of being a physician of color, describes what has been done to develop programs that benefit minority communities, and argues that more such programs are needed.

This study examines the metajournalistic discourse reflected in the use of corpse images from the DusitD2 terror attack in Nairobi, Kenya, in January 2019. Drawing from concepts such as responsibility and resistance ethics, this study explores the viewpoints of Kenyan journalists and bloggers. Situated within qualitative research methodology, the findings suggest that the New York Times’ use of victims’ corpse images reflects a double standard and visual bias, and its defense of the news report can be considered an example of (...) professional posturing. Conversely, sharing of perpetrators’ postmortem pictures produced mixed findings among audiences. While online circulation was blamed on bloggers’ inexperience, it was also seen as a sign of victory for the value of visual evidence. This study also contributes to the scholarship on use of graphic images by drawing on African ethical systems and just war theory. (shrink)

The ethical culture of any organization is not simply a reflection of its mission statement or even its code of conduct. Rather, the real ethics of institutions are often embedded in their reward systems. We suggest how ethics professors can lead students to develop a greater understanding of rewards by providing a review of various forms of organizational rewards. We also offer insights into how professors can compare reward systems in their classes. We conclude by addressing a number of pedagogical (...) approaches that are useful in equipping ethics students with awareness and understanding of reward systems’ power and influence on employee attitudes and behaviors surrounding ethical decision making. (shrink)

Reprint of an article appearing in the Journal of Chinese Philosophy (2011).

In this article we focus on three key precepts shared by Confucianism and the African ethic of Ubuntu: the central value of community, the desirability of ethical partiality, and the idea that we tend to become morally better as we grow older. For each of these broad similarities, there are key differences underlying them, and we discuss those as well as speculate about the reasons for them. Our aim is not to take sides, but we do suggest ways that Ubuntu (...) and Confucianism might have something to learn from each other and perhaps come closer. We hope that our preliminary reflections can inspire further debate and thinking on a theme – dialogues between long-standing and large-scale non-Western traditions – that is bound to increase in importance as non-Western societies play a greater role in the global system and as the search continues for a 'global ethic'. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, an important form of type theory (...) based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s account of discrete spacetime as sets “evolving” over a causal set. We observe that Markopoulou’s proposal may be effectively realized by working within an appropriate frame-valued model of set theory. We go on to show that, within (...) this framework, cover schemes may be used to force certain conditions to prevail in the associated models: for example, rendering the universe timeless, obliterating a given event or forcing it to become the universe’s “beginning”. (shrink)

THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world— explicating such facts presents philosophy with some of its subtlest problems. Let me begin by reminding you of some celebrated past attempts made by philosophers and mathematicians to explicate the nature of mathematics.

This monograph is a follow up to the author's classic text Boolean-Valued Models and Independence Proofs in Set Theory, providing an exposition of some of the most important results in set theory obtained in the 20th century--the independence of the continuum hypothesis and the axiom of choice. Aimed at research students and academics in mathematics, mathematical logic, philosophy, and computer science, the text has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory, and (...) includes recent developments in the field. Numerous exercises, along with the enlarged and entirely updated background material, make this an ideal text for students in logic and set theory. (shrink)

Much more than just an anthology, this survey of humanity's search for the meaning of life includes the latest contributions to the debate, a judicious selection of key canonical essays, and insightful commentary by internationally respected philosophers.

This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...) I have elected to offer detailed technical presentations of just a few important instances. (shrink)

A formal, logical, theory of events is developed and used as the basis for a definition of causation and to provide a pragmatics for causal counterfactuals. The theory begins with with a logical formalization of events as represented in the planner strips. The resulting inertial theories include a common sense law of inertia and their pragmatics is based on the principle of chronological minimization. The theory of events is then developed by removing some of the simplifying assumptions of the strips (...) representation. To begin with the assumption that events always succeed, given that their preconditions are true when they occur, is dropped. Instead a common sense law of change is introduced with the effect that events normally succeed but may fail if some qualification (some unusual or unexpected condition) arises. The pragmatics of the resulting causal theories is provided by refining the pragmatics of inertial theories. In particular, where there is a conflict between change and inertia, preference is given to change. The resulting defeasibility of events makes it possible to remove the strips assumption that only one event occurs at any given time; as conflicting events now result in failure rather than inconsistency. However it is desirable to provide a means for adjudicating between conflicting events. Consequently the theory is extended so that qualification preferences can be specified and the pragmatics are refined accordingly. A further strips assumption is that the effects of an event are invariant. In order to remove this assumption a distinction is drawn between primary events, events of the kind described above, and secondary events. Secondary events are invoked by primary events in appropriate contexts, they are causally dependent on primary events and when successful produce the additional, variant, effects of the primary events which invoke them. As a result, both the context-dependent effects of events (their ramifications) and non-deterministic events (events whose outcome is uncertain) can be represented. The theory of primary and secondary events is then used as the basis for a definition of causation which can be seen as providing a formal account of Mackie's definition of a cause as an inus condition. A major theme of this paper is the conflict between the opposite but complementary principles of change and inertia. For much of the paper the emphasis is on restricting the effects of inertia and favouring change. The resulting imbalance becomes unnatural when considering the combined effects of several events. However the definition of causation makes it possible to state a law of change and inertia, which is used to regulate change and inertia and thus ensure a proper balance between them: while caused change is preferred to inertia, inertia is preferred to uncaused change. The theory is then used to provide a pragmatics for causal counterfactuals, thereby extending the representation of actual causal reasoning to counterfactual causal reasoning. This is illustrated by the use of causal counterfactuals in order to obtain explanations. (shrink)

THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. We shall discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: (...) Plato, Aristotle, Leibniz, and Kant. We conclude with a more detailed discussion of the three “schools” of mathematical philosophy which have emerged in the twentieth century: Logicism, Formalism, and Intuitionism. (shrink)

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...) continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit. (shrink)

Let κ be an infinite cardinal. A κ-complete nonprincipal ultrafilter, or, for short, a κ- ultrafilter on a set A is a (nonempty) family U of subsets of A satisfying (i) S ⊆ U & |S|1 < κ ⇒ ∩S ∈ U (κ-completeness) (ii) X ∈ U & X ⊆ Y ⊆ A ⇒ Y ∈ U, (iii) ∀X ⊆ A [X ∈ U or A – X ∈ U] (iv) {a} ∉ U for any a..

Previous research has demonstrated electroencephalogram (EEG) changes in response to low-odor concentrations, resulting in near-chance detection. Such findings have been taken as evidence for olfaction without awareness. We replicated and extended previous work by examining EEG responses to water-water control, 0.0001, 0.001, 0.01, and 1 ppm isoamyl acetate (IAA) in water paired with water only. Detection was above chance (>50%) for .001 and above, and alpha decreased only to those concentrations, suggesting that EEG changes corresponded to IAA awareness. However, when (...) correct trial EEGs were compared to incorrect trial EEGs during .001 ppm, right posterior/central alpha decreased during incorrect trials and alpha decreased more globally (including frontal sites) during correct trials. These data may not reflect awareness or unawareness per se. Instead, results are discussed regarding activation of perceptual systems in the posterior region during incorrect trials and the activation of frontal action systems during a subset of correct trials. (shrink)

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

In what seems to have been his last paper, Insight and Reflection (1954), Hermann Weyl provides an illuminating sketch of his intellectual development, and describes the principal influences—scientific and philosophical—exerted on him in the course of his career as a mathematician. Of the latter the most important in the earlier stages was Husserl’s phenomenology. In Weyl’s work of 1918-22 we find much evidence of the great influence Husserl’s ideas had on Weyl’s philosophical outlook—one need merely glance through the pages of (...) Space-Time-Matter or The Continuum to see it. Witness, for example, the following passages from the former. (shrink)

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...) will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

In a Vedic hymn, Reality or Being is proclaimed as having “arisen from Nothing”. By contrast, in Jaina cosmology time has no beginning; the universe, uncreated, has always existed.In Plato’s Timaeus the universe is conceived as not having existed eternally, but as having been created at some past time by a demiurge acting on pre-existing substance. We are all familiar with the arresting first line of Genesis.

Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.

Most countries appear to believe that their health system is in a state of semi-crisis with expenditures rising rapidly, with the benefits of many services unknown and with pressure from the public to ensure access to a comprehensive range of services. But whose values should inform decision-making in the health area, and should the influence of different groups vary with the level of decision-making? These questions were put to 54 members of the public and health professionals in eight focus groups. (...) Adopting a different perspective from other studies, participants were not asked if particular groups should be involved in decisions but rather through deliberation and discussion nominated their own potential decision makers. This delivered a clear message that participants saw a legitimate role for a broad range of stakeholders in priority-setting decisions so as to incorporate a diversity of expertise and opinion. Companion themes were the acknowledgment that decisions involve ethical judgments and are not purely technical, that the power of special interest groups (such as clinicians) should be kept in check, and that the process by which decisions are reached is important. The results suggest that qualitative methods of investigation have the potential to improve the legitimacy of policy decisions by contributing to a better understanding of the values of the public and health professionals, and by expanding the range of options available for further research. (shrink)

IN THEIR WELL-KNOWN PAPER, Kochen and Specker (1967) introduce the concept of partial Boolean algebra (pBa) and show that certain (finitely generated) partial Boolean algebras arising in quantum theory fail to possess morphisms to any Boolean algebra (we call such pBa's intractable in the sequel). In this note we begin by discussing partial..

Summarya) Bell tries to formulate more explicitly a notion of “local causality”: correlations between physical events in different space‐time regions should be explicable in terms of physical events in the overlap of the backward light cones. It is shown that ordinary relativistic quantum field theory is not locally causal in this sense, and cannot be embedded in a locally causal theory.b) Clauser, Home and Shimony criticize several steps in Bell's argument that any theory of local “beables” is incompatible (...) with quantum mechanics. It is contended that the Clauser‐Horne derivation of a Bell‐type inequality circumvents his weak steps. The Clauser‐Horne derivation must assume that there are no undetected correlations between choices of controllable variables in two space‐like separated regions. Methodological considerations support this assumption.c) In response to criticism by Shimony, Home, and Clauser, Bell tries to clarify the argument of “The theory of local beables”, and to defend as permissible the hypothesis of free variables.d) Bell's reply to an earlier criticism by Shimony, Clauser, and Home is answered. The convergence of Bell's position towards theirs is noted. (shrink)

Two new intuitionistic equivalents to Zarn’s Lemma are stated and proved.

In this paper the view is developed that classes should not be understood as individuals, but, rather, as "classes as many" of individuals. To correlate classes with individuals "labelling" and "colabelling" functions are introduced and sets identified with a certain subdomain of the classes on which the labelling and colabelling functions are mutually inverse. A minimal axiomatization of the resulting system is formulated and some of its extensions are related to various systems of set theory, including nonwellfounded set theories.

subsets X of A for which ∃x (x ∈ A). The set of functions from A to B is denoted by BA.

This paper is concerned with Wittgenstein's early doctrine of the independence of elementary propositions. Using the notion of a free generator for a logical calculus–a concept we claim was anticipated by Wittgenstein–we show precisely why certain difficulties associated with his doctrine cannot be overcome. We then show that Russell's version of logical atomism–with independent particulars instead of elementary propositions–avoids the same difficulties.

The relation ≤ on R is defined by a ≤ b ⇔ ¬b < a. The open interval (a, b) and closed interval [a, b] are defined as usual, viz. (a, b) = {x: a < x < b} and [a, b] = {x: a ≤ x ≤ b}; similarly for half-open, half-closed, and unbounded intervals.

One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem — employed by Cantor himself in showing that the set of real numbers is uncountable — is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. (...) Classically, these two methods are equivalent. But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e., employs intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. I describe the ways in which these two methods.. (shrink)

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.

The study examined the question of who should make decisions for a National Health Scheme about the allocation of health resources when the health states of beneficiaries could change because of adaptation. Eight semi-structured small group discussions were conducted. Following focus group theory, interviews commenced with general questions followed by transition questions and ended with a ‘focus’ or ‘key’ question. Participants were presented with several scenarios in which patients adapted to their health states. They were then asked their views about (...) the appropriate role of the public, patients and health professionals in making social judgements of quality of life. After discussion and debate, all groups were asked the key question: ‘In light of adaptation, who should evaluate quality of life for the purpose of setting priorities in the allocation of health care?’ In all groups participants presented strong arguments for and against decision making by patients, the public and health professionals. However, most groups thought a representative body which included a range of perspectives should make the relevant judgements. This is at odds with the recommendations in most national pharmaceutical guidelines. The main conclusion of the paper is that health economists and other researchers should explore the possibility of adopting a deliberative, consensus-based approach to evaluating health-related quality of life when such judgements are to be used to inform priority setting in a public system. (shrink)

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

We complete our previous(1, 2) demonstration that there is a family of new solutions to the photon and Dirac equations using spatial and temporal circles and four-vector behaviour of the Dirac bispinor. We analyse one solution for a bound state, which is equivalent to the attractive two-body interaction between a charged point particle and a second, which remains at rest. We show this yields energy and angular momentum eigenvalues that are identical to those found by the usual method of solving (...) of the Dirac equation,(4) including fine structure. We complete our previous derivation(2) of QED from a set of rules for the two-body interaction and generalise these. We show that QED may be decomposed into a two-body interaction at every point in spacetime. (shrink)

If we imagine a chess-board with alternate blue and red squares, then this is something in which the individual red and blue areas allow themselves to be distinguished from each other in juxtaposition, and something similar holds also if we imagine each of the squares divided into four smaller squares also alternating between these two colours. If, however, we were to continue with such divisions until we had exceeded the boundary of noticeability for the individual small squares which result, then (...) it would no longer be possible to apprehend the individual red and blue areas in their respective positions. But would we then see nothing at all? Not in the least; rather we would see the whole chessboard as violet, i.e. apprehend it as something that participates simultaneously in red and blue. (shrink)

In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.

Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In Oppositions and Paradoxes, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, (...) Bell not only motivates abstract conceptual thinking about the paradoxes at issue, he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life and he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or imagine an English professor who time-travels back to 1599 to offer a printing of Hamlet to Shakespeare, so as to help the Bard overcome writer's block and author the play which will centuries later inspire an English professor to travel back in time. These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts. (shrink)

Henry at Work invites readers to rethink how we work today by exploring an aspect of Henry David Thoreau that has often been overlooked: Thoreau the worker. John Kaag and Jonathan van Belle overturn the popular misconception of Thoreau as a navel-gazing recluse who was scornful of work and other mundanities. In fact, Thoreau worked hard—surveying land, running his family’s pencil-making business, writing, lecturing, and building his cabin at Walden Pond—and thought intensely about work in its many dimensions. And (...) his ideas about work have much to teach us in an age of remote work and automation, when many people are reconsidering what kind of working lives they want to have. -/- Through Thoreau, readers will discover a philosophy of work in the office, factory, lumber mill, and grocery store, and reflect on the rhythms of the workday, the joys and risks of resigning oneself to work, the dubious promises of labor-saving technology, and that most vital and eternal of philosophical questions, “How much do I get paid?” In ten chapters, including “Manual Work,” “Machine Work,” and “Meaningless Work,” this personal, urgent, practical, and compassionate book introduces readers to their new favorite coworker: Henry David Thoreau. (shrink)

where f ′ (x) is the derivative of f(x) and A is a quantity whose value depends on both x and δx. Now if it were possible to take δx so small (but not demonstrably identical with 0) that (δx)2 = 0 then (1) would assume the simple form..

In spacetime physics any set C of events—a causal set—is taken to be partially ordered by the relation ≤ of possible causation: for p, q ∈ C, p ≤ q means that q is in p’s future light cone. In her groundbreaking paper The internal description of a causal set: What the universe looks like from the inside, Fotini Markopoulou proposes that the causal structure of spacetime itself be represented by “sets evolving over C” —that is, in essence, by the (...) topos SetC of presheaves on Cop. To enable what she has done to be the more easily expressed within the framework presented here, I will reverse the causal ordering, that is, C will be replaced by Cop, and the latter written as P—which will, moreover, be required to be no more than a preordered set. Specifically, then: P is a set of events preordered by the relation ≤, where p ≤ q is intended to mean that p is in q’s future light cone—that q could be the cause of p, or, equally, that p could be an effect of q. In that case, for each event p, the set p↓ = {q: q ≤ p} may be identified as the causal future of p, or the set of potential effects of p. In requiring that ≤ be no more than a preordering—in dropping, that is, the antisymmetry of ≤—I am, in physical terms, allowing for the possibility that the universe is of Gödelian type, containing closed timelike lines. (shrink)

On the contrary, I find nothing in logistic but shackles. It does not help us at all in the direction of conciseness, far from it; and if it requires 27 equations to establish that 1 is a number, how many will it require to demonstrate a real theorem?