“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and (...) relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

In the sole extended break from his life and varing in this way we can associate a sysied career in England, Alan Turing spent the tem of logic with any constructive ordinal. It may be asked whether such a years 1936–1938 doing graduate work at..

A description and analysis of attitudes to non-human animals in classical and late antique Mediterranean thought.

The notion of autonomy commonly employed in medical ethics literature and practices is inadequate on three fronts: it fails to properly identify nonautonomous actions and choices, it gives a false account of which features of actions and choices makes them autonomous or nonautonomous, and it provides no grounds for the moral requirement to respect autonomy. In this paper I offer a more adequate framework for how to think about autonomy, but this framework does not lend itself to the kinds of (...) practical application assumed in medical ethics. A general problem then arises: the notion of autonomy used in medical ethics is conceptually inadequate, but conceptually adequate notions of autonomy do not have the practical applications that are the central concern of medical ethics. Thus, a revision both of the view of autonomy and the practice of “respect for autonomy” are in order. (shrink)

Biotechnology is the art of manipulating living forms as though they were machines. We have been manipulating, and transforming, living forms since we adopted pastoralist ways-by breeding, domestication, training-but it is only recently that anyone has supposed that we could alter outward forms or behaviour by interfering with the inner mechanisms, the mechanical, biochemical and genetic processes that sustain outward shapes and motions. In the past we could do little more than select parents with desirable characteristics in the hope that (...) they would engender what we wanted, and-once the offspring were there-punish or cajole them to do what we wanted. Biotechnology offers the hope that we could achieve a more secure result by altering the biochemical base on which the characters we desire are founded. It is not clear whether, once the possibility is realized, we could retain our present conceptions of human life. If we could engineer saints, heroes, savants, slaves or psychopaths to order, would any of us be human any more?2. (shrink)

Persons are creatures with a range of personal capacities. Most known to us are also people, though nothing in observation or biological theory demands that all and only people are persons, nor even that persons, any more than people, constitute a natural kind. My aim is to consider what non-personal minds are like. Darwin's Earthworms are sensitive, passionate and, in their degree, intelligent. They may even construct maps, embedded in the world they perceive around them, so as to be able (...) to construct their tunnels. Other creatures may be able to perceive that world as also accessible to other minds, and structure it by locality and temporal relation, without having many personal qualities. Non-personal mind, on both modern materialist and Plotinian grounds, may be the more usual, and the less deluded, sort of mind. (shrink)

Philosophers are usually expected to argue only from premises acceptable to a secular audience, in ways that require no special commitment beyond that to the value of argument itself. As a philosopher, I see no particular reason to deny myself the opportunity to argue from other, more `sectarian', premises, in ways now unfamiliar to an unbelieving nation. In so doing I may (as theistical philosophers often do) sound more traditional than many theologians.

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms (...) and proof processes of that formal system. Gentzen's proof succeeds by utilisingtransfinite inductionto prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to provethe restricted ordinal theorem, that a descending sequence of ordinals, less thanε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction isfinitist. (shrink)