Showing papers in "Studies in logic and the foundations of mathematics in 1971"

Hauptsatz for the Intuitionistic Theory of Iterated Inductive Definitions

Abstract: Publisher Summary This chapter presents a proof theoretical analysis of the intuitionistic theory of generalized inductive definitions iterated an arbitrary finite number of times. Like the Hilbert type systems of first order predicate logic that are used, the theories of single and iterated generalized inductive definitions formulated do not lend themselves immediately to a proof theoretical analysis. Therefore, the chapter reformulates the axioms expressing the principles of definition and proof by generalized induction as rules of inference similar to those introduced by Gentzen in his system of natural deduction for first order predicate logic. As in Gentzen's case, this reformulation leads to a notable systematization that is already in the case of ordinary inductive definitions, the rules corresponding to the axioms that express the principle of definition by induction appearing as introduction rules for the inductively defined predicates, whereas the axioms that express the principle of proof by induction give rise to the corresponding elimination rules. Moreover, the generalized inductive definitions appear as inductive definitions iterated once and the iterated generalized inductive definitions as inductive definitions iterated twice or more.

Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory

Abstract: Publisher Summary This chapter discusses the algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Turing's analysis gives a mathematical analysis of configurational computations. The difference between configuration computations and algorithmic procedures is twofold. Firstly, in configurational computations the objects are symbols, whereas in algorithmic procedures the objects operated on are unrestricted. Secondly, in configurational computations at each stage one has a finite configuration whose size is not restricted before computation. In algorithmic procedures, one fixes beforehand a finite number of registers to hold the objects. The chapter presents mathematical analysis of algorithmic procedures with and without counting. Important features of elementary recursion theory are discussed. Effective definitional schemes, or equivalently, generalized Turing algorithms to generalize elementary recursion theory are applied. Turing's analysis could be improved in relation to configurational computability.

A Survey of Proof Theory II

Abstract: This paper explains recent work in proof theory from a neglected point of view. Proofs and their representations by formal derivations are treated as principal objects of study, not as mere tools for analyzing the consequence relation. Though the paper is principally expository it also contains some material not developed in the literature. In particular, adequacy conditions on criteria for the identity of proofs (in § 1c), and a reformulation of Godeľs second theorem in terms of the notion of canonical representation (in § 1d); the use of normalization, instead of normal form, theorems for a direct proof of closure under Church's rule of the theory of species [in § 2a(ii)] and the useless-ness of bar recursive functionals for (functional) interpretations of systems containing Church's thesis [in §2b(iii)]; the use of ordinal structures in a quantifier-free formulation of transfinite induction (in § 3); the irrelevance of axioms of choice to the explicit realizability of existential theorems both for classical and for Heyting's logical rules (in § 4c) and some new uses of Heyting's rules for analyzing the indefinite cumulative hierarchy of sets (in § 4d); a semantics for equational calculi suitable when terms are interpreted as rules for computation [in Appl. Ia(iii)], and, above all, an analysis of formalist semantics and its relation to realizability interpretations (in App. Ic). A less technical account of the present point of view is in [21].

Qualitative Probability in a Modal Setting

Abstract: Publisher Summary This chapter presents propositional calculi that are obtained by adding to modal propositional calculi a binary operator ≳ carrying the intuitive meaning of “at least as probable as” Extending ideas because of Kripke, a semantic modeling suitable for this kind of logic is presented in the chapter It is shown that filtration theory can be modified in such a way that completeness can be established by using well-known theorem from measurement theory It is assumed that the reader has some familiarity with Kripke type semantics for modal logic and with filtration theory The basic logic for PK is that P is for probability and K is for Kripke, and it is axiomatized by the following axiom system

Infinite Forcing in Model Theory

Abstract: Publisher Summary This chapter presents the application of the forcing concept to model theory. The chapter focuses on the consideration of infinite forcing conditions. The results are obtained that are analogous to several of those obtained previously for finite forcing but also find significant differences. Next, a new link is established between the forcing relation and the classical concepts of Model Theory, and this leads to a kind of compactness theorem for forcing and to the axiomatization of classes of generic structures by infinitary sentences. Some of these results are developed also for finite forcing. To discuss the uses of the approach for finite forcing, it is convenient to assume that the number of relation symbols in K is finite and that there are no function symbols in K.

Translator's Foreword

Abstract: Publisher Summary This chapter presents Mal’cev's study, for algebraic systems generalize models (relational structures), algebras (algebraic structures), and partial algebras. An algebraic system consists of a nonempty base set and a number of basic notions defined on it of four possible kinds––predicate (relation), operation, partial operation, and distinguished element––in practice the last three are special forms of predicates. For the metalanguage Mal’cev employs a naive set theory with the axiom of choice. Mal’cev most frequently uses first-order predicate logic (FOPL) as his formal language, but propositional calculus (PC) and second-order predicate logic (SOPL) also occur.

Ramsey's Theorem does not Hold in Recursive Set Theory

Abstract: Publisher Summary This chapter discusses Ramsey's theorem that does not hold in recursive set theory. The theorem is described, which states that there exists a recursive binary symmetric relation (R) on natural numbers (N) such that no recursively enumerable infinit subset of N is R-homogeneous. The proof of the theorem is based on the existence of two recursively enumerable sets of incomparable degrees of insolvability. The proofs of Ramsey's theorem show that there exist arithmetical R-homogeneous sets for recursive relations R. The existence of an infinite recursively enumerable R -homogeneous set implies that either S 1 is recursive in S 2 or S 2 is recursive in S 1 . There exist recursively enumerable sets of incomparable degrees.

Some Reasons for Generalizing Recursion Theory

Abstract: Publisher Summary This chapter discusses some reasons for generalizing recursion theory (g.r.t.). The chapter corrects two common errors: the first is to suppose that there is just one use of g.r.t., the other is to suppose that there are so many (depending on unspecified purposes). Existing results on various generalizations are referred. The formulation and analysis of the aims of g.r.t. themselves are focussed. The metarecursion theory is discussed. The chapter presents the purposes of g.r.t. that include (a) advancing other parts of logic and mathematics; (b) understanding the mathematical character of ordinary recursion theory; and (c) analysis of a general concept of computation. Work on admissible sets constitutes a refinement because it concerns consequences of only special instances of the replacement scheme. An application of classical recursion theory is that this object is well-suited to the study of infinitary languages.

Axiomatic Recursive Function Theory

Abstract: Publisher Summary This chapter discusses the axiomatic recursive function theory. The results of Wagner-Strong theory are general and they hold for all basic recursive function theories (BRFT's). The chapter focuses on the BRFT and shows that any collection of partial functions satisfying the Kleene enumeration theorem must contain all partial recursive functions. A minimality theorem is obtained for the hyperarithmetic functions. A generalization of the relative categoricity is presented. Relative categoricity is sensitive to that consideration of nonprojectibility. Ordinary recursive function theory and the theory of forcing with finite conditions are discussed. A transparent necessary and sufficient condition on monadic and binary partial functions in order for them to be conservatively extended to a BRFT is given. A related problem considered is to give a transparent necessary and sufficient condition on monadic and binary partial functions together with a distinguished binary partial function for this structure to be conservatively extended to a BRFT.

Diophantine Representation of Recursively Enumerable Predicates

Abstract: Publisher Summary This chapter presents a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients that devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. It is well-known that an algorithm for determining the solvability in integers would yield an algorithm for determining the solvability in positive integers and conversely. Hence, the solvability in positive integers is discussed. Lower-case Latin letters will always be variables whose range is the positive integers. Every recursively enumerable set of positive integers (e.g., the set of all prime numbers) coincides with the set of all positive values of some polynomial with integer coefficients.

Axioms for Computation Theories-First Draft

Abstract: Publisher Summary This chapter presents the first draft of the axioms for computation theories A definition of computation theories is proposed and these structures are studied to get the first step in classifying the known theories One of the difficulties in trying to compare and classify these theories has been the lack of a definition of recursion theory The chapter outlines the basic properties of computation theories This includes proving some of the basic results of ordinary recursion theory, and identifying some of the known theories as computation theories The important concept of finiteness relative to a theory is introduced and studied The approach in the chapter differs from the recursive and bounded definition of metarecursion theory A set is finite relative to a theory if the functional representing quantification over that set is computable in the theory This is a natural approach and allows for direct generalization of the fundamental properties of finite sets

Recursively Mahlo Ordinals and Inductive Definitions

Abstract: Publisher Summary This chapter discusses the recursively Mahlo ordinals and inductive definitions. A characterization of the first recursively Mahlo ordinal and the first recursively hyper-Mahlo ordinalis provided. The large countable ordinals are obtained as the closure ordinals of inductive definitions. Inductive definitions play a central role in hierarchy theory. A classic example is the theory of recursive ordinals. The usual systems of notations for the recursive ordinals are inductively defined by very simple (arithmetic) operations. A version of the Candy theorem on the existence of selection operators is the basic tool. A typical system is defined by an inductive definition consisting of several cases, depending on whether the ordinal reached at a given stage was zero, a successor, notationally singular, and notationally regular.

Normal Form Theorem for Bar Recursive Functions of Finite Type

Abstract: Publisher Summary This chapter presents prove of Normal Form Theorem for the bar recursive functions of finite type. These functions will be represented by the bar recursive terms, which are built up from constants denoting the basic operators of explicit definition, primitive recursion and bar recursion. Rules of conversion will be introduced to express the action of these operators. The Normal Form Theorem asserts that every bar recursive term reduces, by means of a finite sequence of conversions, to a unique normal term. A normal term is one with no convertible subterms. The normal numerical terms (i.e., of type 0) are the numerals, and so, in particular, the theorem asserts that every numerical bar recursive term has a unique and computable value. The proof—that when t reduces to a normal term, the normal term is unique—is elementary. When such a normal term exists, it is called the normal form or value of t. The proof of the Normal Form Theorem will involve a modified version of the notion of convertible term, used to prove the normal form theorem for primitive recursive functions of finite type, with and without bar recursion of type 0.

Notions of Realizability for Intuitionistic Arithmetic and Intuitionistic Arithmetic in all Finite Types

Abstract: Publisher Summary This chapter discusses properties of realizability and modified realizability interpretations for intuitionistic arithmetic HA and intuitionistic arithmetic in all finite types HA ω . The chapter describes the formal systems and presents the model hereditarily recursive operations (HRO) and hereditarily effective operations (HEO) for the intensional and the extensional version of HA ω respectively. The chapter characterizes the axiomatically formulae that can be proved in HA, resp. HA ω to be realizable, resp. modified realizable, resp. Dialectica interpretable. The chapter uses these results and the models HRO, HEO for conservative extension results and consistency results (e.g., consistency of HA with Markov's schema and Church's thesis, HAW is conservative over HA, consistency of certain “axioms of choice” for HRO, HEO) and proves theoretic-closure conditions.

On the decision problem for formulas in which all disjunctions are binary

Abstract: Let Z 1 be the class of closed formulas of the form ∃ a ∀ yKay & A x ∃ u∀yMxuy where Kay and Mxuy are conjunctions of binary disjunctions of signed atomic formulas of the form F αβ or F αβ where F is a binary predicate symbol, and α and β are one of the variables a , x , u and y . We prove in our paper that there is no recursive set which separates the non-satisfiable formulas in Z 1 from those satisfiable in a finite domain.

Hauptsatz for the Theory of Species

Abstract: Publisher Summary This chapter describes that by making use of Girard's idea it is possible to analyze the theory of species by means of the method of computability. It follows from this analysis that every deduction of the theory of species actually reduces to a cut free deduction. The proof is by induction on the length of the deduction. Several cases have to be distinguished depending on how the end formula of the deduction has been inferred. By a finite number of eliminations of main cuts, the theorem reduces to a deduction that consists solely of an assumption, has a cut free main branch with normalizable minor deductions or else ends with an introduction inference. The extension of the treatment of second order logic is given in the chapter.

Predicate-Functor Logics(*)

Abstract: Publisher Summary Bound variable, so characteristic of analysis rather than of algebra, has became central to logic. This new logic has come to constitute even a basic theory of the bound variable; for, all the other desired uses of bound variables can be so paraphrased as to cause the bound variables to figure solely as variables of quantification. For, the logic of quantification excels the old algebras of classes and relations not only in flexibility, but in scope; and so it seems worthwhile to see what it would add up to when couched in just the block like sort of constants and connectives and free variables that are the stock in trade of elementary algebra. Such a translation need have no practical advantages, but it would be an algebraic explanation of the bound variable––an algebraic analysis of analysis. Predicate-functor logic is just adequate to the ordinary logic of quantification and identity. It is shown how to translate closed formulas of ordinary logic into predicate-functor logic; it is convenient to adopt three abbreviations.

Semantic Proof of The Craig Interpolation Theorem for Intuitionistic Logic and Extensions. Part I

Abstract: Publisher Summary This chapter describes the semantic proof of the Craig interpolation theorem for intuitionistic logic and extensions. A semantic proof of a form of Robinson's consistency theorem for the intuitionistic predicate calculus is given. This form is equivalent to the Craig interpolation theorem, for any extension of the intuitionistic predicate calculus. The chapter constructs two partially ordered sets of saturated theories that are isomorphic in a certain sense because in the intuitionistic case, a model is obtained not from a single theory but from a partially ordered set of saturated theories. Two structures, one associated with the Δ-theories and other associated with the ⊜-theories, are constructed; it is proved that these two structures are isomorphic in the common language L 0 ⋂ M 0 .

Infinitary Methods in the Model Theory of Set Theory

Abstract: Publisher Summary This chapter discusses the infinitary methods in the model theory of set theory and also presents the study of the end extensions of models of Zermelo-Fraenkel (ZF) set theory. The theorem of Keisler-Morley, which states that every countable model of ZF has a proper elementary end extension, is focussed. It is shown that if ZF is consistent then there are uncountable models of ZF with no end extensions. The necessary preliminaries are described. All the results are proved using methods and results from infinitary logic. Some of the result on collapsing cardinals used the compactness theorem to prove the theorem of Friedman.

On the GCH At Measurable Cardinals

Abstract: Publisher Summary This chapter focusses on the generalized continuum hypothesis (GCH) at measurable cardinals. The chapter discusses a theorem, in which it is supposed that there is a measurable cardinal, κ , such that at least one of the following holds: (1) 2κ > κ+, (2) Every κ-complete filter over κ can be extended to a κ-complete ultra-filter, and (3) there is a uniform κ-complete ultrafilter over κ+. The statement of the theorem and the proof are formalized within Morse-Kelley set theory with the axiom of choice. It is also assumed that κ is a measurable cardinal satisfying the theorem.

An Intrinsic Characterization of the Hierarchy of Constructible Sets of Integers1)

Abstract: Publisher Summary This chapter provides an intrinsic characterization of the hierarchy of constructible sets of integers. Some of the results are in second-order number theory, and the others are in set theory; standard (incompatible) notations for each are used. The hierarchy of arithmetical degrees is identical with the ramified analytic hierarchy; it is identical with the hyperarithmetic hierarchy where that hierarchy is defined - taking into account the difference between Turing degrees and arithmetical degrees. The division into cases is motivated by the relativized hyperarithmetical hierarchy. Forcing for unlimited statements is defined in terms of forcing for limited statements, and by induction on complexity.

Embeddings into Semigroups with Only a Few Defining Relations

Abstract: Publisher Summary This chapter discusses result that imposes the condition that ℑactually be embedded in the Thue system that is constructed. The two techniques are described in the chapter. In technique 1, there is a uniform construction applicable to any Thue system ℑ. None of the later Thue systems used for embedding will have the simplicity of the present ℑ, taking into account the length of the defining relations, the nature of the mapping that gives the embedding and the explicitness with which the system is given. There is a prove by a rather messy but basically straightforward induction on the number of applications of defining relations of ℑthat are required to transform X into W. In the second technique, a sharper result is obtained than the Corollary of technique 1.

Recursion Theoretic Structure for Relational Systems

Abstract: Publisher Summary This chapter discusses the recursion theoretic structure for relational systems. Given an algebraic structure, there are many ways in which it can be introduced into the study of notions of effectiveness. The chapter discusses some natural methods of doing this. Preliminary discussion on types of algebraic object is presented. The chapter defines various notions of recursiveness and relative recursiveness over an algebraic structure. Maps α : N → E play an important part. A means of importing recursion theory into the structure of E is given. Such maps or rather the equivalence classes of such maps, with respect to recursive interreducibility, become interesting objects in themselves. The chapter develops a complete structure for the set for these equivalence classes and trace relationships between this structure and the nature of E itself.

A Countable Hierarchy for the Superjump

Abstract: Publisher Summary This chapter discusses the countable hierarchy for the superjump. The notion of a recursive functional of finite type was introduced and the usefulness of this concept was illustrated by showing that the hyperarithmetical sets of integers were exactly the sets recursive in the type 2 object 2 E . There can be no countable hierarchy that generates the class of sets of integers recursive in a type 3 object that is at least as strong as 3 E . The chapter reviews Shoenfield's construction and discusses why it works. The superjump is described and the modified countable computation is given, which makes use of Shoenfield's hierarchy. The chapter presents a countable hierarchy of type 2 jumps, and uses its constructed hierarchy to derive some results about recursiveness.

Computability Over the Continuum

Abstract: Publisher Summary This chapter discusses the computability over the continuum Classical descriptive set theory was concerned with classifying sets of real numbers according to the complexity of their definitions from a point of view that considered individual reals as given and thus not subject to analysis Modern hierarchy theory as an outgrowth of recursion theory takes as given only the (potentially) infinite sequence of natural numbers and analyses the complexity of functions and relations of both natural and real numbers The chapter provides several characterizations of the class of hyperprojective functions The abstract theory of hyperprojective relations on arbitrary structures applies directly to any type and leads to the hyper-order n-projective relations

Kreisel's Work on the Philosophy of Mathematics-I. Realism

Abstract: Publisher Summary This chapter reviews Kreisel's work on the philosophy of mathematics. The data of foundations consist of the mathematical experience of the working mathematician; the general problem of foundations is to analyze the experience as a whole. Crude formalism holds that mathematics consists of assertions of the form: a concretely given configuration has been constructed by means of a given mechanical rule. No general statements about such configurations belong to mathematics. The positivist doctrine considers informal derivations either as unreliable or as irrelevant to mathematics. Realism is the assumption that there are basic elements (sets and the membership relation) with the properties assumed in the cumulative hierarchy; that is, the existential assumptions of set theory are valid. Strong realism is committed to the existence of sets, some containing an infinite number of members. Rejection of formalism has led to the development of systems of logic that cannot be represented mechanically, as by a Turing machine.

A Note on Arithmetical Sets of Indiscernibles

Abstract: Publisher Summary This chapter provides a note on arithmetical sets of indiscernibles. It has been proved that there is a recursive partition (of the set N of natural numbers) that possesses no infinite recursively enumerable (Σ 0 1 ) set of indiscernibles; the existence of some infinite set of indiscernibles is the familiar theorem of Ramsey. The chapter presents an entirely different proof based on the theory of retraceable sets. A set is called Σ 0 n , Π 0 n , Δ 0 n if it is definable in prenex normal form with n alternating quantifiers where the first quantifier can be chosen to be respectively existential, universal or both. This general procedure is referred to as the priority method, which could be more accurately replaced by the approximation method or the trial-and error method.

A Simplified Proof for the Unsolvability of the Decision Problem in the Case

Abstract: Publisher Summary This chapter presents a simplified proof for the insolvability of the decision problem in the case ΛVΛ. Using the ideas of Wang and Buchi, the first proof for the insolvability of the decision problem in the case ΛVΛ has been given by Kahr and others. Wang, who invented the domino games, proved that the decision problem for the corner game is unsolvable. Buchi inferred that the decision problem for the case V Λ ΛVΛ is insolvable. The insolvability of the decision problem for the diagonal game leads to the result that the decision problem for ΛVΛ is unsolvable. A minor change consists in replacing the diagonals by the rows of the first quadrant.

The Immortality Problem for Non-Erasing Turing Machines

Abstract: Publisher Summary In this chapter M is considered a Turing machine (TM). An instantaneous description (ID) of M is a triple 〈q,X,n〉 where q∈K, X∈ ∑∞ and n≥1. describes that M is in state q with the read-write head scanning square no. n and that the tape T contains X. M is to stop if M tries to go off the tape at the left end. M is called a non-writing TM if it contains no write-instructions. The immortality problem (IP) associated with a set of TM-s is the problem of deciding, for a given TM in the set, whether or not there exists an immortal ID. It is shown that IP for non-erasing TM-s is decidable, if the tape is allowed to contain ultimately periodic words. If, however, the tape is restricted to contain only a finite number of non-blanks, then the IP for the set of nonerasing TM-s is recursively undecidable (of degree 0").