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.

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)

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)

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)

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.

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 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)

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)

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)

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)

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)

In 1954 H. L. A. Hart wrote that Austin's work has ‘never, since his death … been ignored’. If it never has been completely ignored, interest in it has periodically waxed and waned. The interest definitely waxed in the 1980s. More books were published about Austin in this period than in any other decade since his death in 1859. Although this literature contains discussions of some of the nineteenth-century responses to his work, they are not the focus of it. Certain (...) of the responses remain completely in the dark, while there is more light to shed on at least some of the others. In short, our knowledge of nineteenth-century interpretations of Austin's legal philosophy is very incomplete. (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)

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)

À l'école d'Édith Stein -- À la recherche du geste intérieur -- Révéler la presence -- Anthropologie phénoménologique et théorie de l'éducation dans l'oeuvre d'Édith Stein (Résumé).

Tutte le dottrine tradizionali, da quelle più arcaiche e "primitive" a quelle più evolute e articolate, hanno sempre affermato che l'uomo non è un essere solamente corporeo, sia pur dotato di ingegno e di intelligenza, ma è un essere dalla natura composita, formato da una parte materiale e da una o più componenti invisibili e "sottili" e che in lui risiede un principio immortale di origine divina o comunque soprannaturale. Secondo il modo di vedere tradizionale, l'uomo, per conoscere e realizzare (...) se stesso, deve pertanto prendere coscienza, in primo luogo ed essenzialmente, di questa sua natura di essere composito e immortale, e questa è una di quelle idee universali (per noi forse la più importante) che troviamo diffuse presso tutte le popolazioni del pianeta, anche se con sfumature diverse nelle varie tradizioni..."--Cover flap. (shrink)

Resenha: KARASCH, Mary C. Before Brasília: frontier life in Central Brazil. Albuquerque: University of New Mexico Press, 2016.

In this brief paper, offered as a policy viewpoint, I introduce what I believe to be a novel concept for supporting individual citizen interaction with the U.S. Federal government, termed the individual social account. I explore whether and how the concept might be implemented so as to strengthen the U.S. social safety net and further citizen trust and responsibility in e-government interactions. I illustrate and develop the concept as a platform for reform and suggest and discuss design criteria, surfacing various (...) issues. I conclude with additional thoughts with which to stimulate further research and discussion of the concept and its practicalities. (shrink)

The large percentages of teachers in the United States are white, middle class people who may not understand or identify their own cultural connections within the American culture, let alone elaborate their own ancestral cultural elements. Educators are expected to teach a diverse student population, and preservice teachers should be prepared to plan instruction within a cultural context. Research reflects the importance of teacher candidates identifying and understanding their own culture and elements which define similarities and differences to other cultures. (...) In this study teacher candidates learned an acronym strategy, Cultural L.I.V.E.S., for identifying cultural elements, which could be incorporated for identification of self culture, and expand their knowledge of cultural elements into lesson planning and instruction. Teacher candidates found the strategy useful when attempting to define and remember elements of culture though not everyone recalled all associated elements. More research should be conducted on what preservice teachers think about their own culture and the culture of their prospective students. Teacher candidates must understand the relationships between instructional topics and cultural elements beyond the classroom into the global community. (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)