“Elements of reality” are defined as in the work of Einstein, Podolsky, and Rosen. It is further assumed that the sum or product of twocommuting elements of reality also is an element of reality. An algebra contradiction ensues.

For the sake of simplicity, we adopt the same logical frame as Tarski's in his Wahrheitsbegriff (Wb). There, Tarski is mainly interested in the possibility of explicitely defining truth for an object-language, he does not pay much attention to recursive definitions of truth. We say why. However, recursive definitions have advantages of their own. In particular, we prove the positive theorem: if L is of finite order ≥ 4, then a recursive definition is possible for L (...) in a metalanguage of the same order as L. We indicate how this result could be used for a solution of a generalized version of Frege's paradox. (shrink)

The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe á la Tarski. A set U 0 of codes for small sets is generated inductively at the same time as a function T 0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U 0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive (...) definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen. (shrink)

We introduce a system of simply typed lambda terms and show that a rather comprehensive class of recursion equations on streams or non-wellfounded trees can be solved in our system. Moreover certain conditions are presented which guarantee that the defined functionals are primitive recursive. As a major example we give a co-recursive treatment of Mints’ continuous cut-elimination operator.

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and Shoenfield’s (...) unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

We prove the strong normalization theorem for the natural deduction system for the constructive arithmetic TRDB (the system with Definition by Transfinite Recursion and Bar induction), which was introduced by Yasugi and Hayashi. We also establish the consistency of this system, applying the strong normalization theorem.

The idea of this paper is to approach linear orderings as generalized ordinals and to study how they are made from their initial segments. First we look at how the equality of two linear orderings can be expressed in terms of equality of their initial segments. Then we shall use similar methods to define functions by recursion with respect to the initial segment relation. Our method is based on the use of a game where smaller and smaller initial segments of (...) linear orderings are considered. The length of the game is assumed to exceed that of the descending sequences of elements of the linear orderings considered. By use of such game-theoretical methods we can for example extend the recursive definitions of the operations of sum, product and exponentiation of ordinals in a unique and natural way for arbitrary linear orderings. Extensions coming from direct limits do not satisfy our game-theoretic requirements in general. We also show how our recursive definitions allow very simple constructions for fixed points of functions, giving rise to certain interesting linear orderings. (shrink)

Induction–recursion is a powerful definition method in intuitionistic type theory. It extends inductive definitions and allows us to define all standard sets of Martin-Löf type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the first Mahlo cardinal. In this article we give a new compact formalization of inductive–recursive definitions by modeling them as initial algebras in (...) slice categories. We give generic formation, introduction, elimination, and equality rules generalizing the usual rules of type theory. Moreover, we prove that the elimination and equality rules are equivalent to the principle of the existence of initial algebras for certain endofunctors. We also show the equivalence of the current formulation with the formulation of induction–recursion as a reflection principle given in Dybjer and Setzer 93). Finally, we discuss two type-theoretic analogues of Mahlo cardinals in set theory: an external Mahlo universe which is defined by induction–recursion and captured by our formalization, and an internal Mahlo universe, which goes beyond induction–recursion. We show that the external Mahlo universe, and therefore also the theory of inductive–recursive definitions, have proof-theoretical strength of at least Rathjen's theory KPM. (shrink)

Define a second order tree to be a map between trees. We show that many properties of ordinary trees have analogs for second order trees. In particular, we show that there is a notion of “definition by recursion on a well-founded second order tree” which generalizes “definition by transfinite recursion”. We then use this new notion of definition by recursion to prove an analog of Lusin’s Separation theorem for closure spaces of global sections of a second order (...) tree. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” , these definitions lead to a number of different concepts, which (...) we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

Lange offers an argument that, according to him, “does not show merely that some proofs by mathematical induction are not explanatory. It shows that none are […]”. The aim here is to present a counterexample to his argument.

We refine the definition of II-computability of [12] so that oracles have a “consistent”, but natural, behaviour. We prove a Kleene Normal Form Theorem and closure of semi-recursive relations under ∃1. We also show that in this more inclusive computation theory Post's theorem in the arithmetical hierarchy still holds.

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure - Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if computable is replaced by primitive recursive (p. r., for short), these definitions lead to a number of (...) different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a (...) formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa complexity measure associated withϕ[Blu67], [DW83]. (shrink)

We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a (...) careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be called "computable" rather than "recursive;" we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be "Computability Theory" or simply Computability rather than "Recursive Function Theory.". (shrink)

1. This paper is an informal exposition of a model-theoretic semantics for knowledge and belief set out in full detail else where. Considerations of space and simplicity prevent any recapitulation of tracts of formal definitions. My aim is simply to inform the reader of the alleged existence of one “new direction” in semantics, and to direct him to the original source for its detailed development. I shall explain certain self-imposed limitations on the scope and adequacy conditions of this treatment. Then, (...) in subsequent discussion of the semantical problems which I claim to have treated, I shall try to impart the flavour of the technical methods involved. (shrink)

Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector’s bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector’s bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg’s original proof, however, relies on a detour through Tait’s infinitary terms and the correspondence between ordinal recursion for α < ε₀ and primitive recursion over (...) finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg’s theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Tᵢ then terms built from BR^ℕ,σ for this particular Y are definable in the fragment T_i+max{1,level(σ)}+2. (shrink)

We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in "propositional" fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The "pure," propositional fragment FLR 0 turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of (...) models. (shrink)

We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in "propositional" fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The "pure," propositional fragment FLR$_0$ turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of models.

In [5], Metakides and Nerode introduced the study of recursively enumerable substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] for (...) Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff consideredX, a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets ofωgive rise to r.e. subsets ofX. The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset ofX? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above.In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that givenAany r.e. set andany r.e. open subset ofX, there exists an r.e. open set ℋ which is a subset ofand is dense in and in whichAis coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree ofA. We then go on and establish various results on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2. (shrink)

This article examines central tensions in cybernetics, defined as the study of self-organization, communication, automated feedback in organisms, and other distributed informational networks, from its wartime beginnings to its contemporary adaptations. By examining aspects of both first- and second-order cybernetics, the article introduces an epistemological standpoint that highlights the tension between its definition as a theory of recursion and a theory of control, prediction, and actionability. I begin by examining the historical outcomes of the Macy Conferences to provide a (...) context for cybernetics’ initial development for scientific epistemology, ethics, and socio-political thought. I draw extensively from Norbert Wiener, Heinz von Foerster, Ross Ashby, and Gregory Bateson, key figures of this movement. I then elaborate upon certain premises of cybernetics to further elucidate an intellectual history from which to begin to construct a cybernetic epistemology. I conclude by offering the second-order cybernetic concept of recursivity as a model and method for ethico-epistemological questioning that can account for both the constructive potential and the limitations of cybernetics in science and society. (shrink)

Let N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets $P, Q, R \subseteq N^2$ operations of transposition, composition, and bracketing are defined as follows: $P^\cup = \{\langle x, y\rangle | \langle y, x\rangle \epsilon P\}, PQ = \{\langle x, z\rangle| \exists y\langle x, y\rangle \epsilon P & \langle y, z\rangle \epsilon Q\}$ , and [ P, Q, R] = ∪n ε M(PnQR (...) n). THEOREM. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing. This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and "definition by general recursion." A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [ P∪, A∪ M, R]. The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs. (shrink)

We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result (...) is obtained by constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of $\exists \sum {\begin{array}{*{20}{c}} h \\ 0 \\ \end{array} } $ sentences. We also present an old but previously unpublished proof of the consistency of I∆₀ + ¬ exp + ¬BΣ₁ under the assumption that the size parameter in Lessan's ∆₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of I∆₀ + ¬ exp + ¬BΣ₁ unconditionally has turned out to be so difficult. (shrink)

David Hitchcock, in his recent “Informal Logic and the Concept of Argument”, defends a recursive definition of ‘argument.’ I present and discuss several problems that arise for his definition. I argue that refining Hitchcock’s definition in order to resolve these problems reveals a crucial, but minimally explicated, relation that was, at best, playing an obscured role in the original definition or, at worst, completely absent from the original definition.

This paper argues for a particular semantic decomposition of morphological definiteness. I propose that the meaning of ‘the’ comprises two distinct compositional operations. The first builds a set of witnesses that satisfy the restricting noun phrase. The second tests this set for uniqueness. The motivation for decomposing the denotation of the definite determiner in this way comes from split-scope intervention effects. The two components—the selection of witnesses on the one hand and the counting of witnesses on the other—may take effect (...) at different points in the composition of a constituent, and this has non-trivial semantic consequences when other operators inside the DP take action in between them. In particular, I analyze well-known examples of mutually recursive definite descriptions like ‘the rabbit in the hat’ as examples of definites whose referent-introducing and referent-testing components are interleaved rather than nested. I further demonstrate that this picture leads to a new theory of relative superlative descriptions like ‘the kid who climbed the highest tree’, which explains the previously mysterious role of the definite determiner in licensing such readings. (shrink)