A set is said to be amorphous if it is infinite, but is not the disjoint union of two infinite subsets. Thus amorphous sets can exist only if the axiom of choice is false. We give a general study of the structure which an amorphous set can carry, with the object of eventually obtaining a complete classification. The principal types of amorphous set we distinguish are the following: amorphous sets not of projective type, either bounded or unbounded size of members (...) of partitions of the set into finite pieces), and amorphous sets of projective type, meaning that the set admits a non-degenerate pregeometry, over finite fields either of bounded cardinality or of unbounded cardinality. The hope is that all amorphous sets will be of one of these types. Examples of each sort are constructed, and a reconstruction result for bounded amorphous sets is presented, indicating that the amorphous sets of this kind constructed in the paper are the only possible ones. The final section examines some questions concerned with the resulting cardinal arithmetic. (shrink)

Generic automorphisms of certain homogeneous structures are considered, for instance, the rationals as an ordered set, the countable universal homogeneous partial order, and the random graph. Two of these cases were discussed in [7], where it was shown that there is a generic automorphism of the second in the sense introduced in [10]. In this paper. I study various possible definitions of 'generic' and 'mutually generic', and discuss the existence of mutually generic automorphisms in some cases. In addition, generics in (...) the automorphism group of the rational circular order are considered. (shrink)

It is shown that all the countable superatomic boolean algebras of finite rank have the small index property.

We study a notion of ‘o-amorphous’ which bears the same relationship to ‘o-minimal’ as ‘amorphous’ 191–233) does to ‘strongly minimal’. A linearly ordered set is said to be o-amorphous if its only subsets are finite unions of intervals. This turns out to be a relatively straightforward case, and we can provide a complete ‘classification’, subject to the same provisos as in Truss . The reason is that since o-amorphous is an essentially second-order notion, it corresponds more accurately to 0-categorical (...) o-minimal, and our classification is thus very similar to the one given in 565–592) for that case. More interesting structures arise if we replace ‘interval’ in the definition by ‘convex set’, giving us the class of weakly o-amorphous sets. Here, in fact, there are so many examples that a complete classification seems out of the question. We illustrate some of the structures which these may exhibit, and classify them in certain instances not too far removed from the o-amorphous case. (shrink)

Together, Models and Computability and its sister volume Sets and Proofs will provide readers with a comprehensive guide to the current state of mathematical logic. All the authors are leaders in their fields and are drawn from the invited speakers at 'Logic Colloquium '97' (the major international meeting of the Association of Symbolic Logic). It is expected that the breadth and timeliness of these two volumes will prove an invaluable and unique resource for specialists, post-graduate researchers, and the informed and (...) interested nonspecialist. (shrink)

Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory.

A set is said to be amorphous if it is infinite, but cannot be written as the disjoint union of two infinite sets. The possible structures which an amorphous set can carry were discussed in [5]. Here we study an analogous notion at the next level up, that is to say replacing finite/infinite by countable/uncountable, saying that a set is quasi-amorphous if it is uncountable, but is not the disjoint union of two uncountable sets, and every infinite subset has a (...) countably infinite subset. We use the Fraenkel–Mostowski method to give many examples showing the diverse structures which can arise as quasi-amorphous sets, for instance carrying a projective geometry, or a linear ordering, or both; reconstruction results in the style of [1] are harder to come by in this case. (shrink)

We show that the 'tail' of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

In this paper we describe the well-founded initial segment of the free Heyting algebra ������α on finitely many, α, generators. We give a complete classification of initial sublattices of ������₂ isomorphic to ������₁ (called 'low ladders'), and prove that for 2 < α < ω, the height of the well-founded initial segment of ������α.

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.

A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah . Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S on an infinite cardinal μ are all of the form Sκ = the subgroup consisting of elements whose support has cardinality 20, cƒ 20 < κ, 0 < κ < 20, and κ = 0, we make a further analysis of the first order (...) theory of Sλ/Sκ. introducing many-sorted second order structures , all of whose sorts have cardinality at most 20, and in terms of which we can completely characterize the elementary theory of the groups Sλ/Sκ. (shrink)

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.

It is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel-Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a `generic' extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we (...) also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure. (shrink)

The relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

Tarski [5] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w (X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation |P(X)| = (...) |Y|. (shrink)

0-categorical o-minimal structures were completely described by Pillay and Steinhorn 565–592), and are essentially built up from copies of the rationals as an ordered set by ‘cutting and copying’. Here we investigate the possible structures which an 0-categorical weakly o-minimal set may carry, and find that there are some rather more interesting examples. We show that even here the possibilities are limited. We subdivide our study into the following principal cases: the structure is 1-indiscernible, in which case all possibilities are (...) classified up to binary structure; the structure is 2-indiscernible, classified up to ternary structure; the structure is 3-indiscernible, in which case we show that it is k-indiscernible for every finite k. We also make some remarks about the possible structures of higher arities which an 0-categorical weakly o-minimal structure may carry. (shrink)

This new edition of Law and Medical Ethics continues to chart the ever-widening field that the topics cover. The interplay between the health caring professions and the public during the period intervening since the last edition has, perhaps, been mainly dominated by wide-ranging changes in the administration of the National Health Service and of the professions themselves but these have been paralleled by important developments in medical jurisprudence.

In this paper I describe and analyze an economic situation involving two competitive organizations. I put forth the argument that because of the systemic nature of decision making relative to managing the requirements of utilizing a descriptive equation that determines how many people an economic system can support, that even if all the players in the situation act ethically, the results will still be harmful, and necessarily so, to the system and to many innocent people. I will demonstrate that harming (...) innocent people is required and morally defensible given the choices available. The best one can do is slow down the increase in disutilities and possibly operate the system within a range of acceptable behavior that minimizes the resulting harm to innocent people, even if we cannot entirely remove this harm. In this context ethics becomes the study of the acceptable as opposed to the preferable. (shrink)

_Practical Bioethics_ offers a mix of theory and readings, presented in a format that is succinct and approachable. Each chapter begins and ends with a case study, illustrating the core issues at play and emphasizing the practical nature of the dilemmas arising in medicine. Primary source texts are provided to flesh out the issues, and each of these is carefully edited and presented with interwoven explanatory comments to assist student readers. Throughout, J.K. Miles shows the importance of health-care ethics to (...) us all, whether we be medical providers, patients, or both. (shrink)

Recently, new developments took place in the Dutch debate on the legislation of euthanasia. After a brief account of that debate, the article discusses a new government proposal for legislation in this field, which was submitted to the Dutch parliament in November 1991. This proposal relates not only to euthanasia but also to some other medical decisions concerning the end of life. The author concludes that, for several reasons, it is unsatisfactory.

This article revisits long-standing questions about consent, membership and emigration in Locke’s thought. Commentators such as A John Simmons have argued that Locke opens political membership to both express consenters and some kind of tacit consenters, and not just to the former, as some have suggested. Simmons’s reading seems to render Locke more sensible in that it does not exclude large numbers of people from membership or burden the few members with all the civic duties, and also in that it (...) allows at least tacit-consenting members the right to relocate, while this right is denied to express-consenting members. Against this reading, the article shows, by resolving seemingly conflicting claims in the text, that people become members only by express consent. It also responds to the criticism that the express-consent-only reading would limit membership to a few and so would render Locke’s account implausible from a practical point of view. The article then addresses the purported restriction Locke imposes on express consenters’ right to emigrate, arguing that the restriction concerns the change of membership and not the right to relocate. (shrink)

Originally published in 1926, this book attempts to state 'what has been believed with regard to God's incapacity for suffering'. Mozley charts the development of the doctrine from the Apostolic Fathers through the Reformation to the modern influence of metaphysical philosophy and concludes with six questions intended to prompt further theological discussion on this point. This book will be of value to anyone with an interest in the history of Christian theology.

Recent scholarship on technology-induced ‘conceptual disruption’ has spotlighted the notion of a conceptual gap. Conceptual gaps have also been discussed in scholarship on epistemic injustice, yet up until now these bodies of work have remained disconnected. This article shows that ‘gaps’ of interest to both bodies of literature are closely related, and argues that a joint examination of conceptual disruption and epistemic injustice is fruitful for both fields. I argue that hermeneutical marginalization—a skewed division of hermeneutical resources, which serves to (...) diminish the experiences of marginalized folk—does not only transpire because of conceptual gaps, but also because of two other kinds of conceptual disruption: conceptual overlaps and conceptual misalignments. Hence, there are multiple kinds of conceptual disruption that can be usefully studied through the normative lens of epistemic injustice. Technology can play different roles vis-a-vis epistemic injustices, both as a causal trigger of conceptual disruption, but also as a mediator of hermeneutical resources. Its role is normatively significant, in particular because socially disruptive technologies can have different epistemic implications for different groups: they may amplify the epistemic resources of some groups, while diminishing those of others. (shrink)

The literature on collective action largely ignores the constraints that moral principle places on action-prompting intentions. Here I suggest that neither individualism nor holism can account for the generality of intentional contents demanded by universalizability principles, respect for persons, or proactive altruism. Utilitarian and communitarian ethics are criticized for nominalism with respect to social intentions. The failure of individualism and holism as grounds for moral theory is confirmed by comparing Tuomela's reductivist analysis of we-intentions with Gilbert's analysis of social facts. (...) Tuomela's account founders over intentions to cooperate, and Gilbert's cannot accommodate legitimate authority, vicarious agency, or group structure. (shrink)