Rice's Theorem says that every nontrivia semantic property of programs is undecidable. In this spirit we show the following: Every nontrivia absolute counting property of circuits is UP-hard with respect to polynomial-time Turing reductions. For generators [31] we show a perfect analogue of Rice's Theorem.

This note clarifies an error in the proof of the main theorem of “The Ricean Objection: An Analogue of Rice’s Theorem for First-Order Theories”, Logic Journal of the IGPL, 16(6): 585–590(2008).

We propose here an extension of Rice's Theorem to first-order logic, proven by totally elementary means. If P is any property defined over the collection of all first-order theories and P is non-trivial over the set of finitely axiomatizable theories , then P is undecidable. This not only means that the problem of deciding properties of first-order theories is as hard as the problem of deciding properties about languages accepted by Turing machines, but also offers a general setting (...) for proving several undecidability results in first-order theories. (shrink)

Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, equation image. Other cases are done for all transfinite notations in a very natural, proper subsystem equation image of equation image, where equation image (...) has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in equation image. (shrink)

Answering a question in the reverse mathematics of combinatorial principles, we prove that the thin set theorem for pairs ) implies the diagonally noncomputable set principle over the base axiom system $\mathrm{RCA}_{0}$.

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that equation image. Note that equation image is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate equation image by μ. It is known that whenever A so presents α then A ≤wttα, where ≤wtt (...) denotes weak truth table reducibility, and that the wtt-degrees of presentations form an ideal ℐ in the computably enumerable wtt-degrees. We prove that any such ideal is equation image, and conversely that if ℐ is any nonempty equation image ideal in the computably enumerable wtt-degrees then there is a computable enumerable real α such that ℐ = ℐ. We also prove a kind of Rice Theorem for these ideals, namely that if the index set of such a equation image ideal is not empty or equal to ω then it is equation image-complete. (shrink)

The simulation hypothesis has recently excited renewed interest, especially in the physics and philosophy communities. However, the hypothesis specifically concerns {computers} that simulate physical universes, which means that to properly investigate it we need to couple computer science theory with physics. Here I do this by exploiting the physical Church-Turing thesis. This allows me to introduce a preliminary investigation of some of the computer science theoretic aspects of the simulation hypothesis. In particular, building on Kleene's second recursion theorem, I (...) prove that it is mathematically possible for us to be in a simulation that is being run on a computer \textit{by us}. In such a case, there would be two identical instances of us; the question of which of those is ``really us'' is meaningless. I also show how Rice's theorem provides some interesting impossibility results concerning simulation and self-simulation; briefly describe the philosophical implications of fully homomorphic encryption for (self-)simulation; briefly investigate the graphical structure of universes simulating universes simulating universes, among other issues. I end by describing some of the possible avenues for future research that this preliminary investigation reveals. (shrink)

We argue that Gödel's completeness theorem is equivalent to completability of consistent theories, and Gödel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, (...) and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of them are undecidable. (shrink)

In this short paper, we analyse whether assuming that mathematical objects are “thin” in Linnebo's sense simplifies the epistemology of mathematics. Towards this end, we introduce the notion of transparency and show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparency of some thin objects is a fruitful characteristic mark of abstract mathematics.

We deal with ontological problems concerning basic systems of explicit mathematics, as formalized in Jäger's language of types and names. We prove a generalized inseparability lemma, which implies a form of Rice's theorem for types and a refutation of the strong power type axiom POW + . Next, we show that POW + can already be refuted on the basis of a weak uniform comprehension without complementation, and we present suitable optimal refinements of the remaining results within the (...) weaker theory. (shrink)

In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable (...) numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice. (shrink)

Historically, Ehrenfest’s theorem is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form, and we highlight its generalizations to the relativistic Dirac particle and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.

For Ivor Grattan-Guinness on the occasion of his retirement. The work of Augustus De Morgan on symbolic logic in the mid-nineteenth century is familiar to historians of logic and mathematics alike. What is less well known is his work on probability and, more specifically, the use of probabilistic ideas and methods in his logic. The majority of De Morgan's work on probability was undertaken around 1837???1838, with his earliest publications on logic appearing from 1839, a period which culminated with the (...) publication of his Formal Logic in 1847. This article examines the overlap between his work on probability theory and logic during the earliest period of his interest in both. (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)

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.

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)

Guilt and shame are self-conscious emotions with implications for mental health, social and occupational functioning, and the effectiveness of sports practice. To date, the assessment and role of athlete-specific guilt and shame has been under-researched. Reporting data from 174 junior elite cricketers, the present study utilized exploratory factor analysis in validating the Athletic Perceptions of Performance Scale, assessing three distinct and statistically reliable factors: athletic shame-proneness, guilt-proneness, and no-concern. Conditional process analysis indicated that APPS shame-proneness mediated the relationship between general (...) and athlete-specific distress. While APPS domains of guilt-proneness and no-concern were not significant mediators, they exhibited correlations in the expected direction with indices of psychological distress and well-being. The APPS may assist coaches and support staff identify players who may benefit from targeted interventions to reduce the likelihood of experiencing shame-prone states. (shrink)

Der lebende Organismus stellt einen bestimmten, individuellen Teil des Universums dar, indem er seine Selbständigkeit abgetrennt von dem übrigen Universum erhält, durch den Teilungseffekt, welcher durch die Anordnung seiner Teile bestimmt wird. Es ist anerkannt, dass die Teilung nicht vollständig ist, jedoch eine beschränkte gegenseitige Beziehung zulässt. Es ist auch anerkannt, dass der Grad der Organisation in einer verwandten Reihe sich ändert von den tiefen Verwickelungen der einfachen Zelle zu den mannigfachen Abwandlungen wie sie beim Menschen auftreten. Die Auffassung von (...) effektiver Teilung ist ausserdem auch fest und substantiell auf der Tatsache begründet, die durch das Vorhandensein der Plasmamembran und Grenzflächen kolloidaler Aggregaten gegeben ist. Betont ist die dynamische Natur der charakteristischen Gleichgewichtszustände und es wird aufmerksam gemacht auf die kontinuierliche Bildung von effektiven Teilungen bei verschiedenen Gleichgewichtszuständen. Weitere Untersuchung der Bedeutung dieser Teilungen ergibt viele interessante Folgerungen. Schwierigkeiten finden sich, wenn man eine derartige Beschreibung auf geistige Phänomena ausdehnt. Zwar bestehen grundlegende Übereinstimmungen, aber es ist schwer sie genau zu definieren. Es ist klar, dass eine Synthese dieser Art notwendigerweise unvollständig ist, aber es schien wertvoll den Versuch zu machen, L'organisme vivant est présenté ici comme une partie définie et individualisée de l'univers maintenant son individualité propre au moyen de „l'effet de partition” déterminé par l'arrangement de ses parties. Il est entendu que la partition n'est pas complète mais permet encore des relations réciproques mais limitées. Il est reconnu aussi que le degré d'organisation varie dans le cours d'une série, des complexités profondes de la cellule aux innombrables modifications exhibées par l'organisme humain. Le concept de la partition effective est basé solidement sur l'évidence présentée par la membrane plasmatique et les limites de phases dans les aggrégations colloidales. De plus il est important de noter spécialement la nature dynamique des équilibres caractéristiques et l'on ne saurait manquer d'attirer l'attention sur la formation continue de partitions effectives aux différents niveaux d'équilibre. L'étude ultérieure de la signification de ces partitions présente un grand nombre d'implications intéressantes. Il est à noter que des difficultés sont rencontrées quand on essaie d'inclure en une description de ce genre les phénomènes mentaux et psychiques. Des stabilités fondamentales existent, mais elles sont difficiles à définir avec précision. Il est évident du reste au lecteur qu'une synthèse telle que celle-ci est nécessairement incomplète mais néanmoins intéressante à faire. (shrink)

A proof of Bell’s theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.

Leslie Tharp proves three theorems concerning epistemic and metaphysical modality for conventional modal predicate logic: every truth is a priori equivalent to a necessary truth, every truth is necessarily equivalent to an a priori truth, and every truth is a priori equivalent to a contingent truth. Lloyd Humberstone has shown that these theorems also hold in the modal system Actuality Modal Logic, the logic that results from the addition of the actuality operator to conventional modal logic. We show that Tharp’s (...) theorems fail for the expressively equivalent Subjunctive Modal Logic, the logic that was developed by Kai Wehmeier as an alternative to AML. We then argue that the existence of Tharp’s theorems for AML is due to a faulty interpretation of the notion of necessary truth, a feature that is not shared by SML. The paper concludes with an argument for the thesis that the existence of the distinction between truth at all worlds w and truth at all worlds w from the point of view of w as the actual world is an artifact owing to the interaction of the necessity and actuality operator. (shrink)

In this essay Suzanne Rice examines Aristotle's ideas about virtue, character, and education as elements in an Aristotelian conception of good listening. Rice begins by surveying of several different contexts in which listening typically occurs, using this information to introduce the argument that what should count as “good listening” must be determined in relation to the situation in which listening actually occurs. On this view, Rice concludes, there are no “essential” listening virtues, but rather ways of listening that may be (...) regarded as virtuous in the context of particular concrete circumstances. (shrink)

My examination of Spinoza's arguments for the infinity of extended substance lead to a comparison of his views with the anti-Kantian arguments offered by Moritz Schlick, and finally to some general remarks concerning Spinoza's concept of infinite magnitude, and its limitations from a contemporary perspective.

Gleason's theorem for ������³ says that if f is a nonnegative function on the unit sphere with the property that f(x) + f(y) + f(z) is a fixed constant for each triple x, y, z of mutually orthogonal unit vectors, then f is a quadratic form. We examine the issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason's theorem in light of the recent publication of such a proof.

According to a recent paper by Tim Maudlin, Bell’s theorem has nothing to tell us about realism or the descriptive completeness of quantum mechanics. What it shows is that quantum mechanics is non-local, no more and no less. What I intend to do in this paper is to challenge Maudlin’s assertion about the import of Bell’s proof. There is much that I agree with in the paper; in particular, it does us the valuable service of demonstrating that Einstein’s objections (...) to quantum mechanics have nothing to do with its indeterminism. But I do think that Maudlin’s conclusion is overly cut-and-dried. Quantum mechanics is far from a unified edifice, and what Bell’s theorem shows depends on what version of quantum mechanics you look at. In particular, I’ll try to make the case that there’s an interesting, if ultimately uncompelling anti-realist construal of the import of Bell’s theorem. And I also want to suggest that locality isn’t quite as decisively defeated as Maudlin claims. (shrink)

In a recent paper (Okasha, Mind 120:83–115, 2011), Samir Okasha uses Arrow’s theorem to raise a challenge for the rationality of theory choice. He argues that, as soon as one accepts the plausibility of the assumptions leading to Arrow’s theorem, one is compelled to conclude that there are no adequate theory choice algorithms. Okasha offers a partial way out of this predicament by diagnosing the source of Arrow’s theorem and using his diagnosis to deploy an approach that (...) circumvents it. In this paper I explain why, although Okasha is right to emphasise that Arrow’s result is the effect of an informational problem, he is not right to locate this problem at the level of the informational input of a theory choice rule. Once the informational problem is correctly located, Arrow’s theorem may be dismissed as a problem. (shrink)

Bayes's theorem is a tool for assessing how probable evidence makes some hypothesis. The papers in this volume consider the worth and applicability of the theorem. Richard Swinburne sets out the philosophical issues. Elliott Sober argues that there are other criteria for assessing hypotheses. Colin Howson, Philip Dawid and John Earman consider how the theorem can be used in statistical science, in weighing evidence in criminal trials, and in assessing evidence for the occurrence of miracles. David Miller (...) argues for the worth of the probability calculus as a tool for measuring propensities in nature rather than the strength of evidence. The volume ends with the original paper containing the theorem, presented to the Royal Society in 1763. (shrink)

ObjectiveTo apply a socioecological approach to identify risk and protective factors across levels of the “sports-ecosystem,” which are associated with mental health outcomes among athletes in para-sports and non-para sports. A further aim is to determine whether para athletes have unique risks and protective factor profiles compared to non-para athletes.MethodsA cross-sectional, anonymous online-survey was provided to all categorized athletes aged 16 years and older, registered with the Australian Institute of Sport. Mental health outcomes included mental health symptoms, general psychological distress, (...) risky alcohol consumption and eating disorder risk. Risk and protective factors across multiple levels of the socioecological model, including individual, microsystem, exosystem and macrosystem level factors were assessed via self-report.ResultsA total of 427 elite athletes participated in the study. No significant differences in the rates of mental health problems were observed between para and non-para athletes. Both differences and similarities in risk and protective factor profiles were found across the multiple levels of the sports-ecosystem. Weak evidence was also found to support the hypothesis that certain risk factors, including experiencing two or more adverse life events in the past year, sports related concussion, high self-stigma, inadequate social support and low psychological safety conferred a greater risk for poorer mental health outcomes for para athletes in particular.ConclusionRisk factors occurring across various levels of the sports ecosystem, including individual, interpersonal and organizational level risk factors were found to be associated with a range of poorer mental health outcomes. The association between mental ill-health and certain risk factors, particularly those at the individual and microsystem level, appear to be greater for para athletes. These findings have important implications for policy and mental health service provision in elite sports settings, highlighting the need for more nuanced approaches to subpopulations, and the delivery of mental health interventions across all levels of the sports ecosystem. (shrink)

We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to (...) be the theorem-set. This gives, we believe, an important and hitherto unnoticed connection between mechanism and the use of authorities by human-like epistemic agents. (shrink)

The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.

The value of a company’s ethical reputation has become a focal point for management researchers. We seek to join this conversation and extend the research centered on a firm’s ethical reputation. We accomplish this by shifting our focus away from its impact on external stakeholders to its impact on internal stakeholders. To this end, we rely on signaling theory to explain why a firm’s ethical reputation matters to its employees in an effort to bridge the macro–micro research gap. Across two (...) studies, we propose and demonstrate that a firm’s ethical reputation impacts employee subjective career success in form of career opportunities and work–life balance. Given our signaling theory framework, we also identify and explain when two industry-level characteristics operate as boundary conditions that distort a firm’s ethical reputation signaling properties. Specifically, the results demonstrate that a firm’s ethical reputation is positively related to employees’ perceptions of career opportunities and work–life balance. The results of our studies also demonstrate that the relatively high levels of industry competition and industry regulation weaken the positive impact of a firm’s ethical reputation on career opportunities and work–life balance. Theoretical and practical implications are discussed. (shrink)

What mental and physical distress do actors, camerapersons, and reporters experience when working on reenactments of traumatic moments in history? In Political Camerawork, D. Andy Rice theorizes that the intense feelings produced while creating these performed scenarios, called "simulation documentaries," connect difficult pasts to the present. Building on his background as a nonfiction film director, producer, editor, and cinematographer, Rice analyzes performance techniques to gain insight into the emotional toll of simulation documentaries, including those reliving the Vietnam War, the US (...) military's embodied training in California during the Iraq War, and an annual quadruple lynching reenactment organized by Black civil rights activists in Georgia. Investigating the lasting impact of these productions, Political Camerawork reveals that, by performing a simulation of a traumatic event they didn't directly experience, those involved become carriers of the trauma. (shrink)

Bell's theorem is expounded as an analysis in Bayesian probabilistic inference. Assume that the result of a spin measurement on a spin-1/2 particle is governed by a variable internal to the particle (local, “hidden”), and examine pairs of particles having zero combined angular momentum so that their internal variables are correlated: knowing something about the internal variable of one tells us something about that of the other. By measuring the spin of one particle, we infer something about its internal (...) variable; through the correlation, about the internal variable of the second particle, which may be arbitrarily distant and is by hypothesis unchanged by this measurement (locality); and make (probabilistic) prediction of spin observations on the second particle. Each link in this chain has a counterpart in the Bayesian analysis of the situation. Irrespective of the details of the internal variable description, such prediction is violated by measurements on many particle pairs, so that locality—effectively the only physics invoked—fails. The time ordering of the two measurements is not Lorentz-invariant, implying acausality. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of the statistics of the results imply it has a nonlocal—acausal interpretation; one such, the “transactional” interpretation, is presented to demonstrable advantage, and some misconceptions about quantum theory are pursued. The “unobservability” loophole in photonic Bell experiments is proven to be closed. It is shown that this mechanism cannot be used for signalling; signalling would become possible only if the hidden variables, which we insist must underlie the statistical character of the observations (the alternative is to give up), are uncovered in deviations from quantum predictions. Their reticence is understood as a consequence of their nonlocality: it is not easy to isolate and measure something nonlocal. Once the hidden variables are found, all the problems of quantum field theory and of quantum gravity might melt away. (shrink)

Bell’s theorem admits several interpretations or ‘solutions’, the standard interpretation being ‘indeterminism’, a next one ‘nonlocality’. In this article two further solutions are investigated, termed here ‘superdeterminism’ and ‘supercorrelation’. The former is especially interesting for philosophical reasons, if only because it is always rejected on the basis of extra-physical arguments. The latter, supercorrelation, will be studied here by investigating model systems that can mimic it, namely spin lattices. It is shown that in these systems the Bell inequality can be (...) violated, even if they are local according to usual definitions. Violation of the Bell inequality is retraced to violation of ‘measurement independence’. These results emphasize the importance of studying the premises of the Bell inequality in realistic systems. (shrink)

(2011). Korn and Freidlin's Misunderstanding of the Null Hypothesis Significance Testing Procedure. The American Journal of Bioethics: Vol. 11, No. 3, pp. 15-16.

A Guide to Plato's Republic provides an integral interpretation of the Republic which is accessible even to readers approaching Plato's masterwork for the first time. Written at a level understandable to undergraduates, it is ideal for students and other readers who have little or no background in philosophy or political theory. Rice anticipates their inevitable reactions to the Republic and treats them seriously, opening the way to an appreciation of the complexities of the text without oversimplifying it. While many books (...) on the Republic never stray far from explicating Plato's text, this work contrasts Plato's responses to perennial issues in philosophy and political theory with those of several key subsequent thinkers. It uses engaging examples to show the continuing relevance of Plato's arguments and introduces some basic vocabulary of philosophy and political theory, going beyond terse dictionary definitions by illustrating what technical terms mean in the context of Plato's work. The author's interpretative posture is appreciative but respectfully critical of Plato's vision. Stressing the relationship between Plato's politics and metaphysics, Rice argues that Plato's reluctance to accept the reality and consequences of finitude accounts for much of what many readers find objectionable in his politics. Lively, relatively brief, and designed to provoke discussion in the classroom, A Guide to Plato's Republic is ideal for political theory and introduction to philosophy courses as well as other courses that assign the Republic as a primary text. (shrink)

Excessive alcohol use has adverse effects on the central nervous system (CNS) and can lead to alcohol use disorders (AUDs). Recent studies have suggested that myelin reductions may directly contribute to CNS dysfunctions associated with AUDs. Myelin consists of compact lipid membranes wrapped around axons to provide electrical insulation and trophic support. Regulation of myelin is considered as a new form of neural plasticity due to its profound impacts on the computation of neural networks. In this review, the authors first (...) discuss experimental evidence showing how alcohol exposure causes demyelination in different brain regions, often accompanied by deficits in cognition and emotion. Next, they discuss postulated molecular and cellular mechanisms underlying alcohol's impact on myelin. It is clear that more extensive investigations are needed in this important but underexplored research field in order to gain a better understanding of the myelin‐behavior relationship and to develop new treatment strategies for AUDs. (shrink)

I argue that Spinoza’s account of appetition, and its application to human sexuality, is more original than many commentators suggest; and that it offers resolutions to several puzzles in the philosophy of sex. The paper first situates these puzzles in contemporary debates, offers a detailed analysis of Spinoza’s remarks on love in general and sexual love in particular, and concludes with some of the normative consequences which Spinoza attempts to derive from these.

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms (...) are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma. (shrink)

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 (...) has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles. (shrink)