Our ultimate purpose is to give an axiomatic treatment of recursion theory sufficient to develop the priority method. The direct or abstract approach is to keep in mind as clearly as possible the methods actually used in recursion theory, and then to formulate them explicitly. The indirect or experimental approach is to look first for other mathematical theories which seem similar to recursion theory, to formulate the analogies precisely, and then to search for an axiomatic treatment which covers not only (...) recursion theory but also the analogous theories as particular cases.The first approach is more general because it does not depend on the existence of analogues. A concrete mathematical theory, it seems, need have no such analogues and still be important, as e.g. classical number theory. In such a case, an axiomatic treatment may still be useful for exhibiting the mathematical structure of the theory considered and the assumptions on which it rests. However, it will lack one of the most heavily advertised advantages of the axiomatic method, namely, the “economy of thought” which results from an uniform theory for several different and interesting cases: we cannot hope for this if, by hypothesis, we know of only one particular case. In contrast, the second approach, if successful at all, is bound to achieve such “economy” because we start out with several interesting particular cases. Another possible virtue of the second approach is that of field work over insight: the abstract pattern that we are looking for and hoping to formalize in axioms, may not be evident in any one mathematical theory, but may spring to the eye if one happens to look simultaneously at several theories which happen to realize the pattern. (shrink)

We give picture-covariant formulations of the equations of motion for observables and states such that the Hamiltonian operator is transformed asH-0304;=U(t)HU † (t) under a time-dependent unitary transformationU(t). Next, we consider the explicit and implicit covariance of Heisenberg's equations of motion for observables with respect to general transformations of coordinate operators. Most of our representation is spread out over a number of textbooks and articles, where the subject has been considered with greater or lesser clarity from different points of view.

A type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings). By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (...) (HEO) are easily characterized. (shrink)

If one regards an ordinal number as a generalization of a counting number, then it is natural to begin thinking in terms of computations on sets of ordinal numbers. This is precisely what Takeuti [22] had in mind when he initiated the study of recursive functions on ordinals. Kreisel and Sacks [9] too developed an ordinal recursion theory, called metarecursion theory, which specialized to the initial segment of the ordinals bounded by.The notion of admissibility was introduced by Kripke [11] (...) and Platek [14] and served to generalize metarecursion theory. Kripke called ordinal α admissible if it satisfied certain closure properties of infinitary computations. It was shown that admissibility could be equivalently formulated in terms of the replacement schema of ZF set theory and that α =is an admissible ordinal. The study of a recursion theory on an initial segment of the ordinals bounded by some arbitrary admissible α became known as α-recursion theory.Kripke [10] employed a Gödel numbering scheme to perform an arithmetiza-tion of α -recursion theory and created an analogue to Kleene'sT-predicate of ordinary recursion theory. TheT-predicate then served as the basis for showing that analogues of the major results of unrelativized o.r.t. held in α-recursion theory; namely, the α-Enumeration Theorem,T Theorem, α-Recursion Theorem, and α-Universal Function Theorem. (shrink)

Contents: Recursive Enumerability and the Jump Operator; On the Degrees Less Than 0'; A Simple Set Which Is Not Effectively Simple; The Recursively Enumerable Degrees Are Dense; Metarecursive Sets (with G Kreisel); Post's Problem, Admissible Ordinals and Regularity; On a Theorem of Lachlan and Marlin; A Minimal Hyperdegree (with R O Gandy); Measure-Theoretic Uniformity in Recursion Theory and Set Theory; Forcing with Perfect Closed Sets; Recursion in Objects of Finite Type; The a-Finite Injury Method (with S G Simpson); Remarks (...) Against Foundational Activity; Countable Admissible Ordinals and Hyperdegrees; The 1-Section of a Type n Object; The k-Section of a Type n Object; Post's Problem, Absoluteness and Recursion in Finite Types; Effective Bounds on Morley Rank; On the Number of Countable Models; Post's Problem in E-Recursion; The Limits of E-Recursive Enumerability; Effective Versus Proper Forcing. (shrink)

O presente artigo procede, em primeiro lugar, a um exame das evidências disponíveis referentes à atitude de Wittgenstein em relação ao, bem como conhecimento do, primeiro teorema da incompletude de Gödel, incluindo as suas discussões com Turing, Watson e outros em 1937-1939, e o testemunho posterior de Goodstein e Kreisel Em segundo lugar, o artigo discute a importância filosófica e histórica da atitude de Wittgenstein em relação ao teorema de Gödel e outros teoremas da lógica matemática, contrastando esta atitude (...) com a de, por exemplo, Penrose. Finalmente, a autora responde também a criticas instrutivas feitas por Mark Steiner a um artigo seu publicado em 1995, as quais estabelecem a importância do trabalho semântico de Tarski, quer para o nosso entendimento das observacoes de Wittgenstein. /// This paper presents, first, a survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937-1939, and later testimony of Goodstein and Kreisel Secondly, the article discusses the philosophical and historical importance of Wittgenstein's attitude toward Gödal's and other theorems in mathematical logic, contrasting this attitude with that of, e. g., Penrose. Finally, the author also replies to an instructive criticism of her 1995 paper by Mark Steiner which assesses the importance ofTarskih semantical work, both for our understanding of Wittgenstein's remarks on Godel, and our understanding ofGodeVs theorem itself. (shrink)

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe have asserted the existence of non-empty co-r. e. closed sets devoid of computable points: sets which are even “large” in the sense of positive Lebesgue measure.This leads us to investigate for various classes of computable real subsets whether they always contain a computable point.

A survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937–1939, and later testimony of Goodstein and Kreisel; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g., Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of Tarski's (...) semantical work, both for our understanding of Wittgenstein's remarks on Gödel, and our understanding of Gödel's theorem itself. (shrink)

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic (...) regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski. Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle. (shrink)

La logica matematica di Russell, di K. Gödel.--Che cos'è il problema del continuo di Cantor?, di K. Gödel.--Osservazioni al Convegno su i problemi di matematica per il secondo centenario di Princeton, di K. Gödel.--Matematica e logica, di A. Church.--Osservazioni sulla definizione e sulla natura della matematica, di H.B. Curry.--Sull'infinito, di D. Hilbert.--Il programma di Hilbert, di G. Kreisel.--Fondamenti storici, principi e metodi dell'intuizionismo, di L.E.J. Brouwer.--Disputa, di A. Heyting.--L'intuizionismo in matematica, di A. Heyting.--Verso un nominalismo costruttivo, di N. Goodman (...) e W.V. Quine.--Un mondo di individui, di N. Goodman. (shrink)

An ω-set is a subset of the recursive ordinals whose complement with respect to the recursive ordinals is unbounded and has order type ω. This concept has proved fruitful in the study of sets in relation to metarecursion theory. We prove that the metadegrees of the sets coincide with those of the meta-r.e. ω-sets. We then show that, given any set, a metacomplete set can be found which is weakly metarecursive in it. It then follows that weak relative metarecursiveness is (...) not a transitive relation on the sets, extending a result of G. Driscoll [2, Theorem 3.1]. Coincidentally, we discuss the notions of total and complete regularity. Finally, we solve Post's problem for the transitive closure of weak relative metarecursiveness. We recommend the reader look at pp. 324–328 of the fundamental article [6] of Kreisel and Sacks before proceeding. He will find there a proof of the following very basic fact: a subset of the integers is iff it is metarecursively enumerable (metafinite). (shrink)

This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, i.e. Dialectica and Diller-Nahm interpretation as well as Kreisel's modified realization, together with their Troelstra-style hybrids, are applied to constructive as well as (...) classical systems of arithmetic, analysis, and set theory. They yield relative consistency and conservativity results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations. Prerequisites are: familiarity with classical and intuitionistic predicate logic, basics of computability theory, Gödel's incompleteness theorems. (shrink)

A new method of "minimal" realizability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)--functions f such that HA $\vdash \forall x\exists!yA(x, y)\Rightarrow$ for all m, A(m, f(m)) is true, where A(x, y) may be an arbitrary formula of L(HA) with only x, y free--are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the $ -recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions may always be (...) chosen to be $ -recursive. The method is extended to intuitionistic finite-type arithmetic, HA ω 0 , and elementary analysis. Generalized forms of Kreisel's characterization of the provably recursive functions of PA and of the no-counterexample-interpretation for PA are consequently derived. (shrink)

There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly (...) precisely, it is to be found in the margin of Brouwer's notes for his course on Pointset Theory of 1915/16. The course was repeated in 1916/17 and he must have inserted his first formulation of the continuity principle in the fall of 1916 as new material right at the beginning of the course.In modern language, the principle readswhere α and β range over choice sequences of natural numbers, m and x over natural numbers, and stands for ⟨α, α, …, α⟩, the initial segment of α of length m.An immediate consequence of WC-N is that all full functions are continuous, and, as a corollary, that the continuum is unsplittable [28]. Note that WC-N is incompatible with Church's thesis, [22], section 4.6.After Brouwer asserted WC-N, Troelstra was the first to ask in print for a conceptual motivation, but he remained an exception; most authors followed Brouwer by simply asserting it, cf. [18].Let us note first that in one particular case the principle is obvious indeed, namely in the case of the lawless sequences. The notion of lawless sequence surfaced fairly late in the history of intuitionism. Kreisel introduced it in [17] for metamathematical purposes. There is a letter from Brouwer to Heyting in which the phenomenon also occurs [7]. This is an important and interesting fact since it is the only time that Brouwer made use of a possibility expressly stipulated in, e.g., [5], see below. (shrink)

A collection of 24 out of the 35 papers presented at the Fourth Scandinavian Logic Symposium and First Soviet-Finnish Logic Conference, which took place simultaneously in Finland in 1976. Topics covered are proof theory, set theory, model theory, recursion theory, infinitary languages, generalized quantifiers, truthlikeness, natural language, and "philosophical logic." There is a paper by George Kreisel which discusses an intriguing distinction between the theory of proofs and general proof theory, the latter being the study of the allegedly definitional (...) or essential property of proofs, the former being the study of structural features such as length of proofs and of relations between proofs and other things. There is a paper by Hintikka in which he applies his "game-theoretical semantics" developed elsewhere to quantifiers in natural languages, e.g., the analysis of sentences like "some novel by every novelist is mentioned by every critic". I found the most interesting paper to be the one by Bengt Hansson, which studies the nature of paradoxes by articulating an analogy with mathematical equations, the crucial property about the latter being that they do not always have solutions. Almost all papers are highly technical and thus aimed at experts of the specialties mentioned in the book’s title.—M.A.F. (shrink)

Books reviewed:John Barton and John Muddiman, The Oxford Bible CommentaryLuke Timothy Johnson and William S. Kurz, The Future of Catholic Biblical Scholarship: A Constructive ConversationDavid R. Bauer, An Annotated Guide to Biblical Resources for MinistryDavid Martin, John Orme Mills and W. S. F. Pickering, Sociology and Theology: Alliance and ConflictRichard K. Fenn, The Return of the Primitive: A New Sociological Theory of ReligionJoseph Blenkinsopp, Treasures Old and New: Essays in the Theology of the PentateuchJohn Jarick, 1 ChroniclesMartin Hengel, The Septuagint (...) As Christian Scripture: Its Prehistory and the Problem of Its CanonTricia Gates Brown, Spirit in the Writings of John: Johannine Pneumatology in Social‐Scientific PerspectiveFrederick H. Cryer and Thomas L. Thompson, Qumran between the Old and New TestamentOtto Kaiser, The Old Testament Apocrypha: An IntroductionDavid A. deSilva, Introducing the Apocrypha: Message, Context, and SignificanceMarie E. Isaacs, Reading Hebrews and James: A Literary and Theological CommentaryUwe Michael Lang, Turning Towards the Lord: Orientation in Liturgical PrayerWendy E. Sproston North, The Lazarus Story within the Johannine TraditionAlan Kreider, The Origins of Christendom in the WestBernard Hamilton, The Christian World of the Middle AgesH. E. J. Cowdrey, Pope Gregory VII, 1073–1085G. R. Evans, The Medieval Theologians: An Introduction to Theology in the Medieval PeriodNicholas Vincent, The Holy Blood: King Henry III and the Westminster Blood RelicRichard Dutton, Alison Findlay and Richard Wilson, Theatre and Religion: Lancastrian ShakespeareHoward Kreisel, Prophecy: The History of an Idea in Medieval Jewish PhilosophyHeinz Halm, Shi'ism Carl W. Ernst, Rethinking Islam in the Contemporary WorldMark D. Meyerson and Edward D. English, Christians, Muslims, and Jews in Medieval and Early Modern Spain: Interaction and Cultural ChangeBruce Masters, Christians and Jews in the Ottoman Arab World: The Roots of SectarianismMatthew Dillon, Girls and Women in Classical Greek ReligionJane E. A. Dawson, The Politics of Religion in the Age of Mary, Queen of Scots: The Earl of Argyll and the Struggle for Britain and IrelandSteven Nadler, Spinoza: A LifeDabney Townsend, Hume's Aesthetic Theory: Taste and SentimentImmanuel Kant, Theoretical Philosophy after 1781Felicity McCutcheon, Religion Within the Limits of Language Alone: Wittgenstein on Philosophy and ReligionEmyr Vaughan Thomas, Wittgensteinian Values: Philosophy, Religious Belief and Descriptivist MethodologyTerry Nardin, The Philosophy of Michael OakeshottKirk Ludwig, Donald DavidsonMartin Heidegger, Off the Beaten TrackFrank Lambert, ‘Pedlar in Divinity’: George Whitefield and the Transatlantic RevivalsDeclan Marmion, A Spirituality Of Everyday Faith: A Theological Investigation of the Notion of Spirituality in Karl RahnerHugh Rayment‐Pickard, Impossible God: Derrida's TheologyJoseph Pearce, Old Thunder: A Life of Hilaire Belloc. (shrink)

Let HA be Heyting's arithmetic, and let CS denote the conjunction of Kreisel's axioms for the creative subject: \begin{eqnarray*} {\rm CS}_1.&&\quad \,\forall\, x (\qed_x A \vee \; \neg \qed_x A)\; ,\nn {\rm CS}_2. &&\quad \,\forall\, x (\qed_x A\to A)\; ,\nn {\rm CS}_3^{\rm S}. &&\quad A\to\,\exists\, x \qed_x A\; ,\nn {\rm CS}_4.&&\quad \,\forall\, x\,\forall\, y (\qed_x A & y \ge x\to\qed_y A)\; .\nn \end{eqnarray*} It is shown that the theory HA + CS with the induction schema restricted to arithmetical (i.e. (...) not containing $\qed$ ) formulas is conservative over HA. (shrink)

Interpretation of the classics in political theory seems to go in waves. For a while we had John Locke, the bourgeois thinker. Now we seem to be in a Locke-as-radical-democrat phase. Locke-the-bourgeois had problems of its own, but a radically democratic Locke -- not just the old Locke as liberal democrat but Locke as quasi-Leveller -- strains the interpretative imagination more than most; yet in recent years, several different kinds of argument have been advanced in support of it, both textual (...) and contextual. The most effective argument has proceeded by situating Locke in the context of radical Whig politics in the 1670s and '80s, the struggles over religious toleration and the royal succession, in particular the �Exclusion Crisis� of 1679-81. This contextual argument has been accompanied by various textual interpretations having to do with Locke's conceptions of property, consent, representation, the right of revolution and natural law. Among other things, these are supposed to show that while Locke had nothing explicit to say about the extent of the franchise, the weight of evidence suggests that he would have supported a fairly wide franchise, perhaps even something like the (almost) manhood suffrage advocated by the Levellers (at least according to some, and probably the most convincing, interpretations of their ideas). Most recently, in these pages, Martin Hughes, building on the work of James Tully and Richard Ashcraft in particular, has pushed the argument as far as it can probably go. His argument on taxation and suffrage has provided a motivation here for a wider exploration of Locke's views on representation, consent and the franchise. (shrink)

Imagine a society of fisherfolk, who, in the state of nature, fish on a lake of finite size. Fishing on the lake is characterized by decreasing returns to scale in labor, because the lake's finite size imply that each successive hour of fishing labor is less effective than the previous one, as the remaining fish become less dense in the lake. In the state of nature, the lake is commonly owned: each fishes as much as he pleases, and, we might (...) suppose, calculates his fishing plan by taking the labor of the others as given, as he sees it. Each knows that the distribution of fish will be proportional to labor expended among the fisherfolk: if I fish twice as long as you, I will end up with twice as much fish as you. This is not due to some kind of concern with equity among the fisherfolk; it is a technological fact, implied by the assumption that fishing labor is homogeneous, and all are equally likely to catch a fish in a unit of time. An equilibrium under common ownership can be thought of as a Nash equilibrium of the game where each computes his optimal fishing plan, given the labor of the others and knowing what the consequent distribution of fish would be. (shrink)

In 2005 St Andrews Studies published a volume of essays by Anscombe entitled Human Life, Action and Ethics, followed in 2008 by a second with the title Faith in a Hard Ground. Both books were highly praised. This third volume brings essays on the thought of historical philosophers in which Anscombe engages directly with their ideas and arguments. Many are published here for the first time and the collection provides further testimony to Anscombe's insight and intellectual imagination.

Political decision theory provides a framework for evaluating three approaches to military research and development: offensive weaponry intended for deterrence, the Strategic Defense Initiative and other weaponry intended fordefense, and cutbacks designed to slow the research and development treadmill. Large-scale R&D does not protect against most of the risks facing national security. Nor does an R&D-intensive approach provide the flexibility necessary to adjust military policy in light of rapidly changing international conditions. Considering all factors together, there is a strong theoretical (...) case for placing sharp constraints on military innovation, not just on the quantities of weapons deployed. (shrink)

After recognizing that technologies are socially constructed, questions arise concerning how technologies should be constructed, by what processes, and granting how much influence to whom. Because partisanship, uncertainty, and disagreement are inevitable in trying to answer these questions, reconstructivist scholarship should embrace the desirability of thoughtful partisanship, should focus on strategies for coping intelligently with uncertainties, and should make central the study of social processes for coping with disagreement regarding technoscience and its utilization. That often will entail siding with have‐nots, (...) meaning that reconstructivist scholars often will be opposing the behaviors of government, business, and technoscientific elites. Because reconstructivists normally will be outnumbered, we need to devote more systematic professional attention to setting our collective agenda concerning what is most worth researching. (shrink)

It has frequently been observed that the Prometheus Vinctus shows certain Sophoclean characteristics of rhythm. In order to vary the rhythm of his iambics and to avoid monotony, Sophocles often knits consecutive trimeters closely together by placing at the end of one line some word which looks forward to the next line, and so precludes the reader from stopping for the natural pause after the sixth foot. Sometimes he uses in this way subordinating words which introduce a dependent clause in (...) the next line ; sometimes proclitics or quasi-proclitics. (shrink)

Analyses of the 'food business' expose some of the most fascinating and disturbing characteristics of contemporary capitalism as well as some of the most significant flaws within contemporary academic discourses; deficiencies in diets are the material manifestations of the deficiencies in common analytical and conceptual categories as well as political will. Much of the voluminous recent discourse about sustainable development is similarly flawed. This paper reflects on the connections between the character of contemporary capitalism and allied discourses on globalisation, hunger (...) and sustainable development and argues that these connections require a vigorous ethically informed critique. (shrink)