We study the role the axiom of choice plays in the existence of some special subsets of ℝ and its power set ℘.

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, and (...) that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice , we study partitions of Russell-sets into sets each with exactly n elements , for some integer n. We show that if n is odd, then a Russell-set X has an n -ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell-set X such that |X | is not divisible (...) by any finite cardinal n > 1. (shrink)

We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these (...) models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets ( $${\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}$$ ) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets ( $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ ) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 $${\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}$$ is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which $$2{\mathfrak {m}} = {\mathfrak {m}}$$ for every infinite cardinal number $${\mathfrak {m}}$$. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false. (shrink)

In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...) AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that ACLO ↛ NDS in ZFA set theory. The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that ACWO ↛ NDS in ZF and that “for all infinite cardinals m, m + m = m” ↛ NDS in ZFA. (shrink)

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...) I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with,, where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff (...) spaces are effectively Hausdorff” in. The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in. The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of. The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”. (shrink)

Growing concern has been expressed that we have entered a “post-truth” era in which each of us willfully believes whatever we choose, aided and abetted by alternative and social media that spin alternative realities for boutique consumption. A prime example of the belief in alternative realities is said to be acceptance of “conspiracy theories”—a term that is often used as a pejorative to indict claims of conspiracy that are so obviously absurd that only the unhinged could believe them. The epistemological (...) standard often involved in this indictment, however—the standard of “obvious” falsity—invites subjectivity in its application, because what is obviously false to one person can be common sense to another. This is not just a truism; considerable research suggests that people’s political beliefs, in general, and their acceptance or rejection of conspiracy theories in particular, tends in large part to be determined by partisan, ideological, and other priors. (shrink)

It is now possible to have a child without sexual intercourse. The history of this apparently new statement has been breefly synthetised.The new perspectives open ethical problems for the scientific community and a new social context for single women.

Argues that there can be interaction without breaking physical laws: e.g. by basic psychic forces, or by varying physical constants, or especially by arranging fractal trees of physical causation leading to behavior.

The thought that empathy plays an important role in moral motivation is almost a platitude of contemporary folk psychology. Parallel themes were mooted in German moral philosophy and aesthetics in the 1700s, and versions of the empathy construct remained prominent in continental accounts of moral motivation through the nineteenth century and early twentieth centuries. This chapter elucidates the Empathic Motivation Hypothesis (EMH) and sets out some of the conceptual and empirical challenges it faces. It distinguishes empathic concern from other dimensions (...) of empathy (e.g., resonance, attunement, distress, and non-empathic concern) and examines the merits of EMH as a developmental thesis, focusing on evidence from psychopathology and attachment theory. (shrink)

The main scientific problems of chemical bonding were solved half a century ago, but adequate philosophical understanding of chemical combination is yet to be achieved. Chemists routinely use important terms ("element," "atom," "molecule," "substance") with more than one meaning. This can lead to misunderstandings. Eliminativists claim that what seems to be a baseball breaking a window is merely the action of "atoms, acting in concert." They argue that statues, baseballs, and similar macroscopic things "do not exist." When macroscopic objects like (...) baseballs move, exceedingly large numbers (1025) of microscopic components coordinate their activities. Understanding how this happens requires attention to the interactions that link parts into larger units. Eliminativists say that everything that truly exists has causal relationships in addition to those of its components—"nonredundant causality." This paper holds that if a number of entities interact in such a way that the effect of that collection on test objects is different than it would have been in the absence of the interaction, then identification of that collection as a single composite agent is warranted, for purposes to which that difference is relevant. Ordinary "chemical substances" (both elementary materials such as dihydrogen and compounds such as water) fulfill this version of the requirement of nonredundant causality. Other sorts of chemical coherences, including chemical dissipative structures (e.g., flames), also fulfill that criterion. All these types of coherences qualify as "substances" (as that term is used in philosophy) even though they are not all "chemical substances.". (shrink)

This original and lively book explores Greek ideas about health and disease and their influence on Greek thought. Fundamental issues such as causation and responsibility, purification and pollution, mind-body relations and gender differences, authority and the expert and who can challenge them, reality and appearances, good government, happiness, and good and evil themselves are deeply implicated. Using the evidence not just from Greek medical theory and practice but also from epic, lyric, tragedy, historiography, philosophy, and religion, G. E. R. Lloyd (...) offers the first comprehensive account of the influence of Greek thought about health and disease on the Greek imagination. (shrink)

Galen’s Commentaries on the Hippocratic Epidemics constitute one of the most detailed studies of Hippocratic medicine from Antiquity. The Arabic translation of the Commentaries by Ḥunayn ibn Isḥāq is of crucial importance because it preserves large sections now lost in Greek, and because it helped to establish an Arabic clinical literature. The present contribution investigate the translation of this seminal work into Syriac and Arabic. It provides a first survey of the manuscript tradition, and explores how physicians in the medieval (...) Muslim world drew on it both to teach medicine to students, and to develop a framework for their own clinical research. (shrink)

This review shows the various tools used to implemente the food quality system in food establishments. Presenting the various laws that subsidize this process, establishing criteria and instrumentes for a correct implementation of the quality system in order to offer adequate hygiene processes in the production of safe food, in addition to favoring the maintenance of companies in the consumer market. The implementation of food safety management systems such as Good Manufacturing Practices (GMP), Standard Operating Procedures (SOP) e Hazard Analysis (...) and Critical Control Points (HACCP), in order to guarantee the safety of the food served and portray the organization of the company. It is essential to implement these systems ensuring the quality of the servisse (planning, control and improvement). In order to achieve healthy criteria related to foodstuffs, it is necessary to implement quality programs as prerequisites of the Hazard Analysis System and Critical Control Points in food services. Regulating and inspecting are just a few points that guarantee food security, since food that does not pose a health hazard to consumers is considered safe. (shrink)

The aim of this study is to explore how nurses perceive patient participations of patients over 75 years old undergoing hemodialysis treatment in dialysis units, and of their next of kin. Ten nurses told stories about what happened in the dialysis units. These stories were analyzed with critical discourse analysis. Three discursive practices are found: (1) the nurses’ power and control; (2) sharing power with the patient; and (3) transferring power to the next of kin. The first and the predominant (...) discursive practice can be explained with an ideology of paternalism, in which the nurses used biomedical explanations and the ethical principle of benefice to justify their actions. The second can be explained with an ideology of participation, in which the nurses used ethical narratives as a way to let the patients participate in the treatment. The third seemed to involve autonomous decision-making and the ethical principle of autonomy for the next of kin in the difficult end-of-life decisions. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Interpretation of Plato's Parmenides: Zeno s Paradox and the Theory of Forms R. E. ALLEN PLATO'S Parmenides is divided into three main parts, of uneven length, and distinguished from each other both by their subject matter and their speakers. In the first and briefest part (127d-130a), Socrates offers the Theory of Forms in solution of a problem raised by Zeno. In the second (130a-135d), Parmenides levels a series (...) of objections against Socrates' theory, objections putatively meant as refutation. Then, after an interlude to explain an unfamiliar method of dialectic (135d-137c), that method is applied in the third part by Parmenides, with the help of the young Aristoteles (137c-166c). This final part forms more than two-thirds of the whole. I propose here to examine the first part of the dialogue, the interchange between Zeno and Socrates, and to examine it with special attention to the bearing which questions of dialectical structure and dramatic characterization may have upon its interpretation. It is perhaps here proper to indicate the general view of the dialogue which this discussion will help serve to recommend. Briefly, it is this. The Parmenides is a sustained examination of Plato's Theory of Forms, and the manner of examination is Socratic: the student is presented, not with a body of conclusions, but with a series of arguments whose implications he is left to think through for himself. When he has done so, he will realize that the Theory of Forms is proof against the objections here brought against it: the Parmenides provides no direct support for the thesis, currently popular, that Plato in later life ceased to believe that Forms exist. But though sound as far as it goes, the Theory of Forms does not go far enough: the Parmenides demonstrates that there is a range of problems, bound up with the application of unrestricted predicates, which Forms do not solve but are prey to. A broader theory is required, though a theory which may well contain the Theory of Forms as a part. The Parmenides, whose purpose is critical, leaves unanswered the question of what that theory may be. Zeno's Paradox and the Theoryof Forms (127d-130a) It is Zeno who begins the dialectic of the Parmenides,and fittingly, he begins it with a paradox. If there are many things, the same things must be both [143] 144 HISTORY OF PHILOSOPHY like and unlike. But this is impossible: unlike things cannot be like, nor like things unlike. Therefore, there cannot be many things (127e). Socrates' reply begins with a solution, and ends with a challenge. There exists, alone by itself, a Form of Likeness, and also a Form of its opposite, Unlikeness. The things we call 'many' are both like and unlike; but in this they differ from the Forms in which they partake. For things which are just alike (abr& r& 6uoLa)1 cannot be unlike, nor things just unlike alike. But it is no more surprising that the same things should partake both in Likeness and Unlikeness than it is that a thing should be both one and many by partaking in both Unity and Plurality. Still, Unity itself is not many, nor Plurality one. And so with other Forms: Socrates would be filled with wonder if someone could show that Likeness and Unlikeness, Unity and Plurality, Rest and Motion, and the other Forms, can be combined with (~v'rK~p&vvwOaO and separated from (SLaKotp~0a,) each other. Forms cannot be qualified by their opposite. 2 Socrates' denial is a conjunction. Several translators, including Cornford and Taylor, have rendered ~v3,~p&vvvaOa~ ~as &aKps as though it were a disjunction, 'combined or separated'. The reason for this was no doubt stylistic; but the Greek means, not 'or' but 'and', and logically, the difference is important. Zeno and Parmenides are challenged to prove, not one of two alternatives, but both together. This conclusion rests on more than a Kas In the Sophist, knowing how to separate (&a~ps 253e 3) according to kinds is equivalent to distinguishing (&a~m~a0a~ 253d 1) according to kinds, and both are equivalent to not believing that the same Form is different or a different Form the same (253d 1... (shrink)

In this work Henry Kyburg presents his views on a wide range of philosophical problems associated with the study and practice of science and mathematics. The main structure of the book consists of a presentation of Kyburg's notions of epistemic probability and its use in the scientific enterprise i.e., the effort to modify previously adopted beliefs in the light of experience. Intended for cognitive scientists and people in artificial intelligence as well as for technically oriented philosophers, the book also provides (...) a general overview of the philosophy of science for the non-philosopher by one of the leading authorities in the field. (shrink)

1. Foreword. Man's every act is an act of faith; generally, an unconscious faith; seldom, an examined faith; never, a faith that by taking thought could have been replaced by assured certainties. All this appears whenever we stop to “rationalize” our conduct. Whatever that conduct may have been, it can be made to appear reasonable only if some propositions are taken to be true, others false. Or, perhaps the more experienced and less exacting mind would feel the reasonableness of its (...) actions to have been as well established as might be, if it could show the premises with which its acts were consistent to be more likely true than false; those with whose truth they were inconsistent, more likely false than true. Either way of putting it points to the one moral; namely, that, apart from incalculable chance, there is nothing but one's judgment in the matter of premises to determine all that one's life can come to know of fruition or frustration. Nothing, then, can be more important to any man than to do all he can to assure the soundness of his reasons for the faith that is in him; i.e. to test the weight of evidence supporting the working hypotheses on which he is willing to act. But to have considered the evidence for the hypotheses on which he has acted or is prepared to act, could be practically useful only to one who had in one way or another come by a sound idea of the evidence on which one should act, if one's actions are not to end in disappointment. The result at which a man's thought on this matter of evidence may have arrived, will constitute his theory of evidence; and the task of constructing the soundest possible theory of evidence is the undertaking of philosophy. (shrink)

When we consider the concepts and assumptions of a way of interpreting we are not abstracting ourselves from concrete analytical practice, but are dealing with one dimension of that practice. When a person’s assumptions and concepts change, aspects of his therapeutic work will also change. The philosophical ideal of conceptual clarity means that one strives to be able to recognize how he interprets what is going on—he strives to recognize how he proceeds with the therapeutic process in relation to other (...) options. In that way he has a far better chance of allowing the experience to speak for itself to him, i.e., when he knows something about his own assumptions in relation to a variety of other assumptions, he is able to hear how an occurrence exceeds or differs from his own accustomed ways in which he makes sense of things. (shrink)

The first-century ascetic Jewish philosophers known as the 'Therapeutae', described in Philo's treatise De Vita Contemplativa, have often been considered in comparison with early Christians, the Essenes, and the Dead Sea Scrolls. This study, which includes a new translation of De Vita Contemplativa, focuses particularly on issues of historical method, rhetoric, women, and gender, and comes to new conclusions about the nature of the group and its relationship with the allegorical school of exegesis in Alexandria. Joan E. Taylor argues that (...) the group represents the tip of an iceberg in terms of ascetic practices and allegorical exegesis, and that the women described point to the presence of other Jewish women philosophers in Alexandria in the first century CE. Members of the group were 'extreme allegorizers' in following a distinctive calendar, not maintaining usual Jewish praxis, and concentrating their focus on attaining a trance-like state in which a vision of God's light was experienced. Their special 'feast' was configured in terms of service at a Temple, in which both men and women were priestly attendants of God. (shrink)