The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that "every mapping is strongly extensional", "every sequentially nondiscontinuous mapping is sequentially continuous", and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tsĕitin theorem.

Berger, U., Total sets and objects in domain theory, Annals of Pure and Applied Logic 60 91-117. Total sets and objects generalizing total functions are introduced into the theory of effective domains of Scott and Ersov. Using these notions Kreisel's Density Theorem and the Theorem of Kreisel-Lacombe-Shoenfield are generalized. As an immediate consequence we obtain the well-known continuity of computable functions on the constructive reals as well as a domain-theoretic characterization of the Heriditarily Effective (...) Operations. (shrink)

This paper provides characterisations of the equational theory of the PER model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words, we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model, a version of the Kreisel-Lacombe-Shoenfield theorem (...) allows us to transfer the result from the domain model to the PER model. (shrink)

Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan `open sets are semidecidable properties'. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open. This result has important consequences. Not only follows the classical Rice-Shapiro (...) Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceĭtin-Moschovakis, and a result by Eršov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers. (shrink)

LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. (...) We could, of course, obtain such a contradiction by proving the theorems ofCone by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem inCfrom its proof inC′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.The advantage of the proof may be stated more specifically by arithmetizing the syntax ofCandC′. (shrink)

Hilbert's plan for understanding the concept of infinity required the elimination of non‐finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy of progressively less elementary, but still constructive methods instead of finitist ones . A mathematical proof of this failure requires a definition of « finitist ».—The paper sketches the three principal methods for the syntactic analysis of non‐constructive mathematics, the resulting consistency proofs and constructive interpretations, modelled on Herbrand's theorem, and (...) their mathematical and logical consequences. A characterization of finitist proofs is sketched. A remark on the completeness of the predicate calculus concludes the paper. Throughout open problems and alternative approaches are emphasized.ZusammenfassungHilbert sah das Verstehen des Begriffs des Unendlichen in der Eliminierung nichtfiniter Methoden aus Beweisen finiter Sätze. Das Versagen dieses Unternehmens führt zu einer Hierarchic progressiv weniger elementarer, aber noch konstruktiver Methoden, statt der finiten . Ein Unmöglichkeitsbeweis des ursprünglichen Planes setzt eine Definition von « finit » voraus.—Die drei wichtigsten Methoden der syntaktischen Analyse nichtkonstruktiver mathematischer Theorien, die daraus gewonnenen Widerspruchsfreiheitsbeweise and konstruktiven Interpretationen and deren mathematische and logische Anwendungen werden besprochen. Eine Prazisierung des Begriffs « finit » wird skizziert. Eine Bemerkung zur Vollstandigkeit des engeren Funktionenkalküls beschliesst die Abhandlung.—Oflene Fragen and komplementare Fragestellungen werden betont. (shrink)

IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to translate statements of (...) theorems and proofs in the branch considered into those of some preferred system, where the translation must satisfy certain appropriate conditions (interpretation). The problem is relative to the choice of preferred system, as is Hilbert's consistency problem since he required the consistency to be established by particular methods (finitist ones). A finitist interpretation of classical number theory, which has been published in full detail elsewhere, is here described by means of typical examples. Partial results on analysis (theory of arbitrary functions whose arguments and values are the non-negative integers) are here presented for the first time. One of these results is restricted to functions whose values are bounded; its interest derives from the fact that real numbers may be represented by such functions. It is hoped that diverse general observations and comments, which would bore the specialist, may be of help to the general reader. The specialist may find some points of interest in the last two sections of the main text and in the notes following it. (shrink)

In 1957, Gödel proved that completeness for intuitionistic predicate logic HPL implies forms of Markov's Principle, MP. The result first appeared, with Kreisel's refinements and elaborations, in Kreisel. Featuring large in the Gödel-Kreisel proofs are applications of the axiom of dependent choice, DC. Also in play is a form of Herbrand's Theorem, one allowing a reduction of HPL derivations for negated prenex formulae to derivations of negations of conjunctions of suitable instances. First, we here show how (...) to deduce Gödel's results by alternative means, ones arguably more elementary than those of Kreisel. We avoid DC and Herbrand's Theorem by marshalling simple facts about negative translations and Markov's Rule. Second, the theorems of Gödel and Kreisel are commonly interpreted as demonstrating the unprovability of completeness for HPL, if means of proof are confined within strictly intuitionistic metamathematics. In the closing section, we assay some doubts about such interpretations. (shrink)

We show how Kreisel's representation theorem for sets in the analytical hierarchy can be generalized to sets defined by positive induction and use this to estimate the complexity of constructions in the theory of domains with totality.

A set S of degrees is said to be an initial segment if c ≤ d ∈ S→-c∈S. Shoenfield has shown that if P is the lattice of all subsets of a finite set then there is an initial segment of degrees isomorphic to P. Rosenstein [2] (independently) proved the same to hold of the lattice of all finite subsets of a countable set. We shall show that “countable set” may be replaced by “set of cardinality at most that (...) of the continuum.” This result is also an extension of [3, Corollary 2 to Theorem 15], which states that there is a sublattice of degrees isomorphic to the lattice of all finite subsets of 2N. (A sublattice of degrees is a subset closed under ∪ and ∩; an initial segment closed under ∪ is necessarily a sublattice, but not conversely.)It seems worth noting that the proof of the present result was preceded in time by our proof [4] of the analogous theorem for hyperdegrees, and is in fact an adaptation of that proof. Thus the present work has been influenced much more directly by the Gandy-Sacks forcing construction of a minimal hyperdegree [1] than by previous work on initial segments of degrees. (shrink)

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only (...) use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of (...) L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of th.. (shrink)

We formulate an abstract version of the finite injury method in the form of the Baire category theorem. The theorem has the following corollaries: The Friedberg-Muchnik pair of recursively enumerable degrees, the Sacks splitting theorem, the existence of a minimal degree below 0′ and the Shoenfield jump theorem.

By generalizing Kreisel’s proof of the Second Incompleteness Theorem of G¨odel I extract a general principle which can also be used for otherpurely model-theoretical proofs of that theorem.

It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] $\prec$ [b] $\prec$ [a] there exists [c] ∈ R/M such that [c] $\prec$ [a] and [a] = [b] ∨ [c].

Within a weak subsystem of second-order arithmetic , that is -conservative over , we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.

The generalization properties of algebraically closed fields $ACF_p$ of characteristic $p > 0$ and $ACF_0$ of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that $ACF_p$ admits finite term bases, and $ACF_0$ admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some $k$ , $A(1 + \cdots + 1)$ ( $n$ 1's) is provable in $k$ steps, then $(\forall x)A(x)$ is provable.

Introduction. Il est bien connu que la correspondance de Curry-Howard permet d'associer un programme, sous la forme d'un λ-terme, à toute preuve intuitionniste, formalisée dans le calcul des prédicats du second ordre. Cette correspondance a été étendue, assez récemment, à la logique classique moyennant une extension convenable du λ-calcul. Chaque théorème formalisé en logique du second ordre correspond donc à une spécification de programme.Il se pose alors le problème, en général tout à fait non trivial, de trouver la spécification associée (...) à un théorème donné; autrement dit, de déterminer le comportement opérationnel commun aux λ-termes associés aux diverses démonstrations formelles du théorème considéré.Cette question est résolue ici pour le théorème de complétude de la logique classique.La première étape consiste à formaliser convenablement ce théorème en logique du second ordre. Ce travail est fait complètement dans la section 1. Il a comme sous-produit, peut-être inattendu, de montrer que ce théorème est prouvable en logique intuitionniste du second ordre. Ceci, toutefois, à condition d'introduire une légère variante de la notion de modèle, en admettant un modèle supplémentaire trivial, où toute formule est satisfaite.On notera, à ce sujet, que des preuves intuitionnistes du théorème de complétude de la logique intuitionniste, utilisant des variantes de la notion de modele de Kripke, ont été données par H. Friedman [7] et W. Veldman [8]. On remarquera également qu'un argument de G. Kreisel [2] montre que le théorème de complétude habituel de la logique intuitionniste n'a pas de preuve intuitionniste. (shrink)

The two‐dimensional version of the classical Mycielski theorem says that for every comeager or conull set there exists a perfect set such that. We consider a strengthening of this theorem by replacing a perfect square with a rectangle, where A and B are bodies of some types of trees with. In particular, we show that for every comeager Gδ set there exist a Miller tree and a uniformly perfect tree such that and that cannot be a Miller tree. (...) In the case of measure we show that for every subset F of of full measure there exists a uniformly perfect tree such that and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions from which we derive nonstandard proofs of Mycielski‐like theorems via the Shoenfield Absoluteness Theorem. (shrink)

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (...) (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

Shoenfield’s completeness theorem states that every true first order arithmetical sentence has a recursive \-proof encodable by using recursive applications of the \-rule. For a suitable encoding of Gentzen style \-proofs, we show that Shoenfield’s completeness theorem applies to cut free \-proofs encodable by using primitive recursive applications of the \-rule. We also show that the set of codes of \-proofs, whether it is based on recursive or primitive recursive applications of the \-rule, is \ complete. (...) The same \ completeness results apply to codes of cut free \-proofs. (shrink)

Abstract.Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary first-order predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PL-proofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the β-normal-form of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended (...) language, the terms are ordinary PL-terms. In contrast to approaches to Herbrand’s theorem based on cut elimination orɛ-elimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrand-term extraction tool. (shrink)

From the point of view of non-classical logics, Heyting's implication is the smallest implication for which the deduction theorem holds. This book studies properties of logical systems having some of the classical connectives and implication in the neighbourhood of Heyt ing's implication. I have not included anything on entailment, al though it belongs to this neighbourhood, mainly because of the appearance of the Anderson-Belnap book on entailment. In the later chapters of this book, I have included material that might (...) be of interest to the intuitionist mathematician. Originally, I intended to include more material in that spirit but I decided against it. There is no coherent body of material to include that builds naturally on the present book. There are some serious results on topological models, second order Beth and Kripke models, theories of types, etc., but it would require further research to be able to present a general theory, possibly using sheaves. That would have postponed pUblication for too long. I would like to dedicate this book to my colleagues, Professors G. Kreisel, M.O. Rabin and D. Scott. I have benefited greatly from Professor Kreisel's criticism and suggestions. Professor Rabin's fun damental results on decidability and undecidability provided the powerful tools used in obtaining the majority of the results reported in this book. Professor Scott's approach to non-classical logics and especially his analysis of the Scott consequence relation makes it possible to present Heyting's logic as a beautiful, integral part of non-classical logics. (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.

In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it is not (...) known if these implications reverse, BGS also showed that those theorems imply finite choice (in some cases, with additional induction assumptions). This motivated us to study the proof-theoretic strength of finite choice. Using a variant of Steel forcing with tagged trees, we show that finite choice is not provable from the $\Delta ^1_1$ -comprehension scheme (even over $\omega $ -models). We also show that finite choice is a consequence of the arithmetic Bolzano–Weierstrass theorem (introduced by Friedman and studied by Conidis), assuming $\Sigma ^1_1$ -induction. Our results were used by BGS to show that several theorems in graph theory cannot be proved using $\Delta ^1_1$ -comprehension. Our results also strengthen results of Conidis. (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)

Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded middle and which involve only two quantifier alternations. By contrast, we show that the proof mining of classical Nonstandard Analysis has a (...) very large scope. In particular, we will observe that this scope includes any theorem of pure Nonstandard Analysis, where 'pure' means that only nonstandard definitions are used. In this note, we survey results in analysis, computability theory, and Reverse Mathematics. (shrink)

In [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals Φ A of order type 0 which realize the Herbrand normal form A H of A. Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination were found. These proofs (...) however do not carry out the no-counterexample interpretation as a local proof interpretation and don't respect the modus ponens on the level of the no-counterexample interpretation of formulas A and A → B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (δ) and--at least not constructively-- (γ) which are part of the definition of an 'interpretation of a formal system' as formulated in [15]. In this paper we determine the complexity of the no-counterexample interpretation of the modus ponens rule for (i) PA-provable sentences, (ii) for arbitrary sentences A, B ∈ L(PA) uniformly in functionals satisfying the no-counterexample interpretation of (prenex normal forms of) A and A → B, and (iii) for arbitrary A, B ∈ L(PA) pointwise in given α( 0 ) -recursive functionals satisfying the no-counterexample interpretation of A and A → B. This yields in particular perspicuous proofs of new uniform versions of the conditions (γ), (δ). Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15], [16]. In particular we show that the no-counterexample interpretation of PA n by α( n (ω))-recursive functionals (n ≥ 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (δ). (shrink)

Anderson and Csima :245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the (...) Jump Theorem holds for the bounded jump: do we have A≤bTB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \le _{bT}B$$\end{document} if and only if Ab≤1Bb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^b \le _1 B^b$$\end{document}? We show the forward direction holds but not the reverse. (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)

The long‐standing issue of Wittgenstein's controversial remarks on Gödel's Theorem has recently heated up in a number of different and interesting directions [, , ]. In their , Juliet Floyd and Hilary Putnam purport to argue that Wittgenstein's‘notorious’ “Contains a philosophical claim of great interest,” namely, “if one assumed. that →P is provable in Russell's system one should… give up the “translation” of P by the English sentence ‘P is not provable’,” because if ωP is provable in PM, PM (...) is ω ‐inconsistent, and if PM is ω‐inconsistent, we cannot translate ‘P’as ’P is not provable in PM’because the predicate‘NaturalNo.’in ‘P’“cannot be…interpreted” as “x is a natural number.” Though Floyd and Putnam do not clearly distinguish the two tasks, they also argue for “The Floyd‐Putnam Thesis,” namely, that in the 1930's Wittgenstein had a particular understanding of Gödel's First Incompleteness Theorem. In this paper, I endeavour to show, first, that the most natural and most defensible interpretation of Wittgenstein's and the rest of is incompatible with the Floyd‐Putnam attribution and, second, that evidence from Wittgenstein's Nachla strongly indicates that the Floyd‐ Putnam attribution and the Floyd‐Putnam Thesis are false. By way of this examination, we shall see that despite a failure to properly understand Gödel's proof—perhaps because, as Kreisel says, Wittgenstein did not read Gödel's 1931 paper prior to 1942‐Wittgenstein's 1937–38, 1941 and 1944 remarks indicate that Gödel's result makes no sense from Wittgenstein's own perspective. (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)

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)

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).