This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

In their 2002 seminal paper Hauser, Chomsky and Fitch hypothesize that recursion is the only human-specific and language-specific mechanism of the faculty of language. While debate focused primarily on the meaning of recursion in the hypothesis and on the human-specific and syntax-specific character of recursion, the present work focuses on the claim that recursion is language-specific. We argue that there are recursive structures in the domain of motor intentionality by way of extending John R. Searle’s analysis of intentional action. (...) We then discuss evidence from cognitive science and neuroscience supporting the claim that motor-intentional recursion is language-independent and suggest some explanatory hypotheses: linguistic recursion is embodied in sensory-motor processing; linguistic and motor-intentional recursions are distinct and mutually independent mechanisms. Finally, we propose some reflections about the epistemic status of HCF as presenting an empirically falsifiable hypothesis, and on the possibility of testing recursion in different cognitive domains. (shrink)

In this paper we continue our work of Kalantari and Welch . There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend themselves to effectivization. While we studied recursive points in that paper, here, we present two useful classes of recursive functions on topological spaces, apply them to the reals, and find precise accounting for the nature of the properties of some examples that exist in the literature. (...) We end with a construction of a recursive function on a small subset of the unit interval which is strongly nonextendible. (shrink)

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)

We investigate systems of ordinary differential equations with a parameter. We show that under suitable assumptions on the systems the solutions are computable in the sense of recursive analysis. As an application we give a complete characterization of the recursively enumerable sets using Fourier coefficients of recursive analytic functions that are generated by differential equations and elementary operations.

In this paper a new method, elimination of Skolem functions for monotone formulas, is developed which makes it possible to determine precisely the arithmetical strength of instances of various non-constructive function existence principles. This is achieved by reducing the use of such instances in a given proof to instances of certain arithmetical principles. Our framework are systems ${\cal T}^{\omega} :={\rm G}_n{\rm A}^{\omega} +{\rm AC}$ -qf $+\Delta$ , where (G $_n$ A $^{\omega})_{n \in {\Bbb N}}$ is a hierarchy of (weak) subsystems (...) of arithmetic in all finite types (introduced in [14]), AC-qf is the schema of quantifier-free choice in all types and $\Delta$ is a set of certain analytical principles which e.g. includes the binary König's lemma. We apply this method to show that the arithmetical closures of single instances of $\Pi^0_1$ -comprehension and -choice contribute to the growth of extractable bounds from proofs relative to ${\cal T}^\omega$ only by a primitive recursive functional in the sense of Kleene. In subsequent papers these results are widely generalized and the method is used to determine the arithmetical content of single sequences of instances of the Bolzano-Weierstraß principle for bounded sequences in ${\Bbb R}^d$ , the Ascoli-lemma and others. (shrink)

In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

A development of a constructive fragment of analysis: "constructive" in the strong sense that instead of, say, Cauchy sequences, it deals only with recursive sequences of rationals which can be recursively shown to converge. Analogues of classical subjects such as continuity and differentiability are explored in detail. The book presupposes familiarity with both classical analysis and the theory of recursive functions.—R. H. T.

Mauldin [15] proved that there is an analytic set, which cannot be represented by $B\cup X$ for some Borel set B and a subset X of a $\boldsymbol{\Sigma }^0_2$ -null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated $\sigma $ -ideals having approximation property under the assumption that every real is constructible, answering Mauldin’s question raised in [15].

Language sciences have long maintained a close and supposedly necessary coupling between the infinite productivity of the human language faculty and recursive grammars. Because of the formal equivalence between recursion and non-recursive iteration; recursion, in the technical sense, is never a necessary component of a generative grammar. Contrary to some assertions this equivalence extends to both center-embedded relative clauses and hierarchical parse trees. Inspection of language usage suggests that recursive rule components in fact contribute very little, and likely nothing significant, (...) to linguistic creativity. Further than this, if the productivity of human language is considered as not rigidly bound, but not infinite, then the need for any sort of iteration in generative grammars vanishes and can be replaced with a Non-iterative Explicit Alternatives Rule grammar. The knock-on effects of dispensing with recursive grammar components are: that language diversity can simply be based on rule and lexicon combinatorics with no potentially infinite dimensions derived from recursive or iterative components of rules; the oddity of a vast, multiply-infinite competence set of ‘grammatical but unacceptable’ productions is gone; and the development of a language faculty based on rules that eschew iterative rule components avoids any need for explaining ‘special’ mechanisms. On the broader front of searching for the mechanisms of mind, our analysis can be similarly applied to the proposals for a recursive basis for mind as an explanation for humanity’s great leap forward. (shrink)

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal (...) class='Hi'>analysis. (shrink)

We develop the concepts of recursively nowhere dense sets and sets that are recursively of first category and study closed sets of points in light of Baire's Category Theorem. Our theorems are primarily concerned with exdomains of recursive quantum functions and hence with avoidable points . An avoidance function is a recursive function which can be used to expel avoidable points from domains of recursive quantum functions. We define an avoidable set of points to be an arbitrary subset of the (...) avoidable points of a single avoidance function, and we study an effective union of such sets which we call a piecemeal avoidable set. We note that each of ‘the set of recursive points’ and ‘the set of avoidable points’ is of first category but not recursively of first category. We show an exdomain exists which is recursively nowhere dense, as well as one that is nowhere dense but not recursively nowhere dense. After establishing that every exdomain is recursively of first category, we prove that given any fixed exdomain, there is another exdomain, which while dense in the underlying space, is disjoint from the fixed exdomain. Finally, we show how to build a recursive sequence of recursive quantum functions that have mutually disjoint, dense exdomains. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination which characterises (...) this move. (shrink)

A metric space is said to be locally non-compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non-compact iff it is without isolated points.The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable (...) sequence that is eventually computably bounded away from every computable element of the space. (shrink)

Let the chain antichain principle (CAC) be the statement that each partial order on $\mathbb{N}$ possesses an infinite chain or an infinite antichain. Chong, Slaman, and Yang recently proved using forcing over nonstandard models of arithmetic that CAC is $\Pi^1_1$-conservative over $\text{RCA}_0+\Pi^0_1\text{-CP}$ and so in particular that CAC does not imply $\Sigma^0_2$-induction. We provide here a different purely syntactical and constructive proof of the statement that CAC (even together with WKL) does not imply $\Sigma^0_2$-induction. In detail we show using a (...) refinement of Howard's ordinal analysis of bar recursion that $\text{WKL}_0^\omega+\text{CAC}$ is $\Pi^0_2$-conservative over PRA and that one can extract primitive recursive realizers for such statements. Moreover, our proof is finitary in the sense of Hilbert's program. CAC implies that every sequence of $\mathbb{R}$ has a monotone subsequence. This Bolzano-Weierstraß}-like principle is commonly used in proofs. Our result makes it possible to extract primitive recursive terms from such proofs. We also discuss the Erdős-Moser principle, which—taken together with CAC—is equivalent to $\text{RT}^2_2$. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of-analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to-formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated-comprehension, e.g.,-comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory came (...) into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2is still something to be sought. However, for reasons that cannot be explained here,-comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2in the foreseeable future.Roughly speaking,ordinally informativeproof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding (...) sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel. (shrink)

This book is an investigation of algorithmic contingency and an elucidation of the contemporary situation that we are living in: the regular arrival of algorithmic catastrophes on a global scale. Through a historical analysis of philosophy, computation and media, this book proposes a renewed relation between nature and technics.

This paper presents an in-depth analysis and research on the spatial distribution pattern of the urban landscape in the Central Plains digital landscape form and proposes an optimization scheme. Based on the basic theories of systematics and complexity, this paper analyzes the self-similar characteristics of urban morphology, establishes the concept of schema, and constructs a multiscale and multilevel morphological map research framework by drawing on the “planar pattern” morphological analysis method of the school and the “matrix, patch, and (...) corridor” spatial expression model of landscape ecology. The framework of morphological map research at multiple scales has been established, and the theory and method of describing, understanding, judging, and analyzing morphological evolution have been formed. Cities have self-similarity at different scales, and urban evolution is a process of recursion from small-scale hierarchy to large-scale hierarchy, and hierarchy is a phenomenon presented by the natural evolution of cities. After any morphological process is completed, it can only be transformed into the other two ways, so the static morphological description can be transformed into dynamic morphological process analysis. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of -analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to -formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated -comprehension, e.g., -comprehension. The details will be laid out in (...) [28].Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2 is still something to be sought. However, for reasons that cannot be explained here, -comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2 in the foreseeable future.Roughly speaking, ordinally informative proof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

This paper introduces an extension A of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α : R(α)} is continuous there. The domains of continuity for A coincide with the stable relations (those equivalent in A to their double negations), while every relation (...) R(α) is equivalent in A to ∃αA(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy). The logic of A is intuitionistic. The axioms of A include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, A is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that A preserves the constructive sense of "or" and "there exists.". (shrink)

We show how two iterated products of selection functions can both be used in conjunction with systemTto interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over systemTto Spector’s bar recursion, whereas the other isT-equivalent to modified bar recursion. Modified bar recursion itself is shown to arise directly from the iteration of a different binary product of ‘skewed’ selection functions. Iterations of the dependent binary products are also considered (...) but in all cases are shown to beT-equivalent to the iteration of the simple products. (shrink)

It is shown that in any model of constructive mathematics in which a certain omniscience principle is false, for strongly extensional functions on an interval the distinction between sequentially continuous and regulated disappears. It follows, without the use of Markov's Principle, that any recursive function of bounded variation on a bounded closed interval is recursively sequentially continuous.

Five recursively axiomatizable theories extending Kleene's intuitionistic theory FIM of numbers and numbertheoretic sequences are introduced and shown to be consistent, by a modified relative realizability interpretation which verifies that every sequence classically defined by a Π11 formula is unavoidable and that no sequence can fail to be classically Δ11. The analytical form of Markov's Principle fails under the interpretation. The notion of strongly inadmissible rule of inference is introduced, with examples.

In 1962 Umberto Eco published his Opera aperta. Forma e indeterminazione nelle poetiche contemporanee, in which he dealt with the televised space and its influence on the development of plot in contemporary narratives. The analysis of the aesthetic of television led him to highlight the exclusive capacity of television to transmit events in real time: Live TV.Eco affirms in particular that through the editing in Live TV, the role of choice completely changes in comparison to what happens in the (...) editing phase within the cinematographic narrative. In Live TV, choice becomes a proper composition, a form of narrative, a means of unifying in a discursive way a set of isolated images within the framework of a wider set of events taking place at the same time and intersecting one another. This impromptu narrative brings with it the use of some recursive forms, which represent an important narrative tool enabling a weaving of the narrative space. Eco identified those recursive forms in some jazz figures such as the riff, the recursive form of which allows for the creation of an organic composition full of improvisation. A few years later, the importance of figures of this kind was also noted by Deleuze-Guattari, who posited that ritornello has an important function of organizing the exterior space, or chaotic space. Based on these considerations, this paper deals with the phenomenon of repetition, recursivity, and patterns, to bring these concepts into the construction of narrative spaces. The central point of the paper is that the birth of Live TV started a phenomenon of exteriorization of the inner data of the narrative construction. This phenomenon of externalization makes possible the understanding of data through spatial terms, and one starts to use recursive formulas to navigate it. (shrink)

This paper is concerned with the event-based state and fault estimation problem for a class of linear discrete systems with randomly occurring faults and missing measurements. Different from the static event-based transmission mechanism with a constant threshold, a dynamic event-based mechanism is exploited here to regulate the threshold parameter, thus further reducing the amount of data transmission. Some mutually independent Bernoulli random variables are used to characterize the phenomena of ROFs and missing measurements. In order to simultaneously estimate the system (...) state and the fault signals, the main attention of this paper is paid to the design of recursive filter; for example, for all DETM, ROFs, and missing measurements, an upper bound for the estimation error covariance is ensured and the relevant filter gain matrix is designed by minimizing the obtained upper bound. Moreover, the rigorous mathematical analysis is carried out for the exponential boundedness of the estimation error. It is clear that the developed algorithms are dependent on the threshold parameters and the upper bound together with the probabilities of missing measurements and ROFs. Finally, a numerical example is provided to indicate the effectiveness of the presented estimation schemes. (shrink)

We use the multiple price list method and a recursive expected utility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over gains and (...) are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of-analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to-formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated-comprehension, e.g.,-comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory came (...) into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2is still something to be sought. However, for reasons that cannot be explained here,-comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2in the foreseeable future.Roughly speaking,ordinally informativeproof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of-analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to-formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated-comprehension, e.g.,-comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory came (...) into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2is still something to be sought. However, for reasons that cannot be explained here,-comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2in the foreseeable future.Roughly speaking,ordinally informativeproof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

The class of recursive functions over the reals, denoted by , was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class was put in relation with the class of functions generated by the General Purpose Analogue Computer of Claude Shannon; classical digital computation was embedded in several ways into the new model of computation; restrictions of were proved to represent different classes of recursive functions, e.g., recursive, primitive recursive (...) and elementary functions, and structures such as the Ritchie and the Grzergorczyk hierarchies.The class of real recursive functions was then stratified in a natural way, and and the analytic hierarchy were recently recognised as two faces of the same mathematical concept.In this new article, we bring a strong foundational support to the Real Recursive Function Theory, rooted in Mathematical Analysis, in a way that the reader can easily recognise both its intrinsic mathematical beauty and its extreme simplicity. The new paradigm is now robust and smooth enough to be taught. To achieve such a result some concepts had to change and some new results were added. (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)

Several extensions of Gödel's system TT with new forms of recursion have been designed for the purpose of giving a computational interpretation to classical analysis. One can organise many of these extensions into two groups: those based on bar recursion , which include Spector's original bar recursion, modified bar recursion and the more recent products of selections functions, or those based on open recursion which in particular include the symmetric Berardi–Bezem–Coquand functional. We relate these two groups by showing that (...) both open recursion and the BBC functional are primitive recursively equivalent to a variant of modified bar recursion. Our results, in combination with existing research, essentially complete the classification up to primitive recursive equivalence of those extensions of system TT used to give a direct computational interpretation to choice principles. (shrink)

This study aimed to validate career decision ambiguity tolerance scale-Korean form applicable to a Korean sample. In study 1, 17 items from the original 18-item career decision ambiguity tolerance scale were valid based on IRT. In study 2, using the confirmatory factor analysis, we showed that excluding item 4 from the original scale is better than including it in the three factors model. Given the results of study 1 and 2, the constructs in the 17-item career decision ambiguity tolerance (...) scale-Korean form were valid. In study 3, career decision ambiguity tolerance positively predicted career decision-making self-efficacy, career indecision, and career adaptability respectively, after controlling for calling and career search self-efficacy. Thus, the incremental validity of the career decision ambiguity tolerance scale-Korean form was ensured. In study 4, the reliability of the scale was retained as the test-retest (conducted over a 4-week period) demonstrated adequate results. (shrink)

Matthean scholars have predominantly viewed Jesus’ healing ministry through the lens of ‘fulfillment of prophecy’, which connects his healings to David the shepherd and the fulfilment of the covenant, the restoration of the covenant people, and the establishment of the new covenant. This interpretation has largely emerged from an analysis of Jesus’ healing ministry as a singular event. However, it is necessary to revisit previous studies that have posited that the stories of Jesus’ healings were arranged in a larger (...) context and theological perspective, rather than as a disjointed list of individual events. In particular, the healing events in Matthew 8–9 appear to have been arranged with the intention of teaching discipleship, indicating that the healing stories in the Gospel of Matthew, while scattered, are part of a cohesive pattern of theological themes. These themes include the forgiveness of sins, compassion and mercy, healing through faith, and confession of Jesus as the Messiah. The author of Matthew employs this pattern to systematically present a theological perspective that acknowledges Jesus as divine and confesses him as the Messiah. Contribution: This study aimed to analyse the pattern of these theological themes as they appear in Jesus’ healing narrative. The repetition of these themes serves to establish Jesus’ divinity and reinforce the confession of him as the Messiah. Furthermore, the pattern of theological themes in the healing narrative suggests that it was constructed during a period of transition for the Matthean community, as they moved from Judaism to Christianity. The pattern highlights the community’s confession of Jesus as God and their efforts to solidify their faith. (shrink)

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.

Thomas Hurka, in his book Virtue, Vice, and Value, and elsewhere, develops a recursive analysis of higher-order pleasures and pains. The account leads Hurka to some potentially controversial conclusions. For instance, Hurka argues on its basis that some states are both good and evil and also that the view he calls the conditionality view is false. In this paper, I argue that Hurka’s formulation of the recursive account is unusual and inelegant, and that Hurka reaches his conclusions only because (...) of its peculiar aspects. I provide an alternative recursive account which is similar in spirit to Hurka’s but which is theoretically elegant and does not entail those conclusions. (shrink)

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

We study concepts of decidability for subsets of Euclidean spaces ℝk within the framework of approximate computability . A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability (...) of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces. (shrink)