Results for ' ultrapowers of the real numbers'

943 found
Order:
  1.  29
    Sequential real number computation and recursive relations.J. Raymundo Marcial-Romero & M. Andrew Moshier - 2008 - Mathematical Logic Quarterly 54 (5):492-507.
    In the first author's thesis [10], a sequential language, LRT, for real number computation is investigated. That thesis includes a proof that all polynomials are programmable, but that work comes short of giving a complete characterization of the expressive power of the language even for first-order functions. The technical problem is that LRT is non-deterministic. So a natural characterization of its expressive power should be in terms of relations rather than in terms of functions. In [2], Brattka examines a (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  2. Facts, values, and 'real'numbers.Sophia Mihic, Stephen G. Engelmann & Elizabeth Rose Wingrove - 2005 - In George Steinmetz (ed.), The politics of method in the human sciences: positivism and its epistemological others. Durham: Duke University Press.
     
    Export citation  
     
    Bookmark  
  3.  29
    Real numbers, continued fractions and complexity classes.Salah Labhalla & Henri Lombardi - 1990 - Annals of Pure and Applied Logic 50 (1):1-28.
    We study some representations of real numbers. We compare these representations, on the one hand from the viewpoint of recursive functionals, and of complexity on the other hand.The impossibility of obtaining some functions as recursive functionals is, in general, easy. This impossibility may often be explicited in terms of complexity: - existence of a sequence of low complexity whose image is not a recursive sequence, - existence of objects of low complexity but whose images have arbitrarily high time- (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  4. Real numbers: From computable to random.Cristian S. Calude - 2001 - Studia Philosophica 1.
    A real is computable if it is the limit of a computable, increasing, computably converging sequence of rational...
     
    Export citation  
     
    Bookmark  
  5. How real are real numbers?Gregory Chaitin - 2011 - Manuscrito 34 (1):115-141.
    We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  6.  24
    Real numbers and other completions.Fred Richman - 2008 - Mathematical Logic Quarterly 54 (1):98-108.
    A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields.
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  7.  19
    Monotonically Computable Real Numbers.Robert Rettinger, Xizhong Zheng, Romain Gengler & Burchard von Braunmühl - 2002 - Mathematical Logic Quarterly 48 (3):459-479.
    Area number x is called k-monotonically computable , for constant k > 0, if there is a computable sequence n ∈ ℕ of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k · |x — xn| ≥ |x — xm| for any m > n and x is monotonically computable if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence s ∈ ℕ (...)
    Direct download  
     
    Export citation  
     
    Bookmark   3 citations  
  8.  87
    H‐monotonically computable real numbers.Xizhong Zheng, Robert Rettinger & George Barmpalias - 2005 - Mathematical Logic Quarterly 51 (2):157-170.
    Let h : ℕ → ℚ be a computable function. A real number x is called h-monotonically computable if there is a computable sequence of rational numbers which converges to x h-monotonically in the sense that h|x – xn| ≥ |x – xm| for all n andm > n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h : ℕ → ℚ, we show that (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  9.  52
    Completeness for systems including real numbers.W. Balzer & M. Reiter - 1989 - Studia Logica 48 (1):67 - 75.
    The usual completeness theorem for first-order logic is extended in order to allow for a natural incorporation of real analysis. Essentially, this is achieved by building in the set of real numbers into the structures for the language, and by adjusting other semantical notions accordingly. We use many-sorted languages so that the resulting formal systems are general enough for axiomatic treatments of empirical theories without recourse to elements of set theory which are difficult to interprete empirically. Thus (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  10.  30
    Some Remarks on Real Numbers Induced by First-Order Spectra.Sune Kristian Jakobsen & Jakob Grue Simonsen - 2016 - Notre Dame Journal of Formal Logic 57 (3):355-368.
    The spectrum of a first-order sentence is the set of natural numbers occurring as the cardinalities of finite models of the sentence. In a recent survey, Durand et al. introduce a new class of real numbers, the spectral reals, induced by spectra and pose two open problems associated to this class. In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey. Specifically, we prove that (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  11.  27
    An induction principle over real numbers.Assia Mahboubi - 2017 - Archive for Mathematical Logic 56 (1-2):43-49.
    We give a constructive proof of the open induction principle on real numbers, using bar induction and enumerative open sets. We comment the algorithmic content of this result.
    Direct download  
     
    Export citation  
     
    Bookmark  
  12. Kant and real numbers.Mark van Atten - unknown
    Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a potentially infinite sequence. In this paper I discuss the systematical reasons why in Kant's philosophy this identification is impossible.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  13.  59
    Differential calculus and nilpotent real numbers.Anders Kock - 2003 - Bulletin of Symbolic Logic 9 (2):225-230.
    Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid (...)
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  14. Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?Nicolas Gisin - 2019 - Erkenntnis 86 (6):1469-1481.
    It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  15. Real numbers, quantities, and measurement.Bob Hale - 2002 - Philosophia Mathematica 10 (3):304-323.
    Defining the real numbers by abstraction as ratios of quantities gives prominence to then- applications in just the way that Frege thought we should. But if all the reals are to be obtained in this way, it is necessary to presuppose a rich domain of quantities of a land we cannot reasonably assume to be exemplified by any physical or other empirically measurable quantities. In consequence, an explanation of the applications of the reals, defined in this way, must (...)
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  16.  30
    A Real Number Structure that is Effectively Categorical.Peter Hertling - 1999 - Mathematical Logic Quarterly 45 (2):147-182.
    On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real (...) in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well-known non-appropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers. (shrink)
    Direct download  
     
    Export citation  
     
    Bookmark   6 citations  
  17. Primitive recursive real numbers.Qingliang Chen, Kaile Kaile & Xizhong Zheng - 2007 - Mathematical Logic Quarterly 53 (4):365-380.
    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 (...)
     
    Export citation  
     
    Bookmark  
  18.  87
    Stevin Numbers and Reality.Karin Usadi Katz & Mikhail G. Katz - 2012 - Foundations of Science 17 (2):109-123.
    We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   13 citations  
  19.  21
    Numbers and proofs.Reg Allenby - 1997 - New York: Copublished in North, South, and Central America by John Wiley & Sons.
    'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  20. Galileo Goes to Jail and Other Myths about Science and Religion.Ronald L. Numbers - 2009 - Journal of the History of Biology 42 (4):823-824.
     
    Export citation  
     
    Bookmark   32 citations  
  21. Darwinism Comes to America.Ronald L. Numbers - 1999 - Journal of the History of Biology 32 (2):415-417.
  22.  42
    Primitive recursive real numbers.Qingliang Chen, Kaile Su & Xizhong Zheng - 2007 - Mathematical Logic Quarterly 53 (4‐5):365-380.
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  23.  38
    Clarifying creationism: five common myths.Ronald L. Numbers - 2011 - History and Philosophy of the Life Sciences 33 (1):129-139.
  24.  37
    Recursive real numbers.A. H. Lachlan - 1963 - Journal of Symbolic Logic 28 (1):1-16.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  25. A C.E. Real That Cannot Be SW-Computed by Any Ω Number.George Barmpalias & Andrew E. M. Lewis - 2006 - Notre Dame Journal of Formal Logic 47 (2):197-209.
    The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  26. When Science & Christianity Meet.David C. Lindberg & Ronald L. Numbers - 2005 - Journal of the History of Biology 38 (1):182-184.
     
    Export citation  
     
    Bookmark   2 citations  
  27.  92
    Numbers and Propositions Versus Nominalists: Yellow Cards for Salmon & Soames. [REVIEW]Rafal Urbaniak - 2012 - Erkenntnis 77 (3):381-397.
    Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  28.  47
    Ω in number theory.Toby Ord - 2007 - In Christian Calude (ed.), Randomness & Complexity, from Leibniz to Chaitin. World Scientific Pub Co. pp. 161-173.
    We present a new method for expressing Chaitin’s random real, Ω, through Diophantine equations. Where Chaitin’s method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  29. (1 other version)On virtual classes and real numbers.R. M. Martin - 1950 - Journal of Symbolic Logic 15 (2):131-134.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark  
  30.  50
    Set-theoretical basis for real numbers.Hao Wang - 1950 - Journal of Symbolic Logic 15 (4):241-247.
  31.  19
    Creation by Natural Law: Laplace's Nebular Hypothesis in American Thought.Ronald L. Numbers - 1977
    Belief in the divine origin of the universe began to wane most markedly in the nineteenth century, when scientific accounts of creation by natural law arose to challenge traditional religious doctrines. Most of the credit - or blame - for the victory of naturalism has generally gone to Charles Darwin and the biologists who formulated theories of organic evolution. Darwinism undoubtedly played the major role, but the supporting parts played by naturalistic cosmogonies should also be acknowledged. Chief among these was (...)
    Direct download  
     
    Export citation  
     
    Bookmark   5 citations  
  32.  46
    Introduction to proof through number theory.Bennett Chow - 2023 - Providence, Rhode Island, USA: American Mathematical Society.
    Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  33.  39
    Creationism, intelligent design, and modern biology.Ronald L. Numbers - 2010 - In Denis R. Alexander & Ronald L. Numbers (eds.), Biology and Ideology From Descartes to Dawkins. London: University of Chicago Press.
    Charles Darwin's Origin of Species, published in 1859, was a revolutionary attempt “to overthrow the dogma of separate creations,” a declaration that provoked different reactions among the religious, ranging from mild enthusiasm to anger. Christians sympathetic to Darwin's effort sought to make Darwinism appear compatible with their religious beliefs. Two of Darwin's most prominent defenders in the United States were the Calvinists Asa Gray, a Harvard botanist, and George Frederick Wright, a cleric-geologist. Gray, who long favored a “special origination” in (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  34.  26
    Provably recursive real numbers.William J. Collins - 1978 - Notre Dame Journal of Formal Logic 19 (4):513-522.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  35.  41
    Normal Numbers and Limit Computable Cantor Series.Achilles Beros & Konstantinos Beros - 2017 - Notre Dame Journal of Formal Logic 58 (2):215-220.
    Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a Δn+10 basic sequence with respect to which no Δn0 real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark  
  36.  8
    Science and Christianity in Pulpit and Pew.Ronald L. Numbers - 2007 - Oxford University Press USA.
    As past president of both the History of Science Society and the American Society of Church History, Ronald L. Numbers is uniquely qualified to assess the historical relations between science and Christianity. In this collection of his most recent essays, he moves beyond the clichés of conflict and harmony to explore the tangled web of historical interactions involving scientific and religious beliefs. In his lead essay he offers an unprecedented overview of the history of science and Christianity from the (...)
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  37.  90
    Collimation processes in quantum mechanics interpreted in quantum real numbers.John Vincent Corbett & Thomas Durt - 2009 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 40 (1):68-83.
  38.  44
    Wang Hao. Set-theoretical basis for real numbers.J. Barkley Rosser - 1951 - Journal of Symbolic Logic 16 (3):216-216.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  39.  52
    R. S. Lehman. On primitive recursive real numbers. Fundamenta mathematicae, vol. 49 , pp. 105–118.Paul Axt - 1962 - Journal of Symbolic Logic 27 (2):245-246.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  40.  37
    Martin R. M.. On virtual classes and real numbers.J. C. C. McKinsey - 1951 - Journal of Symbolic Logic 16 (1):64-64.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  41.  51
    Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective.Yulia A. Tyumeneva, Galina Larina, Ekaterina Alexandrova, Melissa DeWolf, Miriam Bassok & Keith J. Holyoak - 2018 - Thinking and Reasoning 24 (2):198-220.
    Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities are aligned with addition, whereas certain functional relations between entities are aligned with division. Similarly, discreteness vs. continuity of quantities is aligned with different formats for rational numbers. These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  42. Biology and Ideology From Descartes to Dawkins.Denis R. Alexander & Ronald L. Numbers (eds.) - 2010 - London: University of Chicago Press.
    Over the course of human history, the sciences, and biology in particular, have often been manipulated to cause immense human suffering. For example, biology has been used to justify eugenic programs, forced sterilization, human experimentation, and death camps—all in an attempt to support notions of racial superiority. By investigating the past, the contributors to _Biology and Ideology from Descartes to Dawkins_ hope to better prepare us to discern ideological abuse of science when it occurs in the future. Denis R. Alexander (...)
    Direct download  
     
    Export citation  
     
    Bookmark   7 citations  
  43.  28
    Stål Anderaa (Oslo), A Traktenbrot inseparability theorem for groups. Peter Dybjer (G öteborg), Normalization by Yoneda embedding (joint work with D. Cubric and PJ Scott). Abbas Edalat (Imperial College), Dynamical systems, measures, fractals, and exact real number arithmetic via domain theory. [REVIEW]Anita Feferman, Solomon Feferman, Robert Goldblatt, Yuri Gurevich, Klaus Grue, Sven Ove Hansson, Lauri Hella, Robert K. Meyer & Petri Mäenpää - 1997 - Bulletin of Symbolic Logic 3 (4).
  44.  41
    N. A. Šanin. Konstruktivnyé véščéstvénnyé ĉisla i konstruktivnyé funkcional'nyé prostranstva (Constructive real numbers and constructive functional spaces). Problémy konstrukticnogo napravléniá v matématik, 2 (Konstruktivnyj matématičéskij analiz), Sbornik rabot, edited by N. A. Šanin, Trudy Matématičéskogo Instituta imèni V. A. Stéklova, vol. 67Izdatél′stvo Akadémii Nauk SSSR, Moscow and Leningrad1962, pp. 15–294. [REVIEW]E. M. Fels - 1969 - Journal of Symbolic Logic 34 (2):290-292.
  45. Ethics without numbers.Jacob Nebel - 2024 - Philosophy and Phenomenological Research 108 (2):289-319.
    This paper develops and explores a new framework for theorizing about the measurement and aggregation of well-being. It is a qualitative variation on the framework of social welfare functionals developed by Amartya Sen. In Sen’s framework, a social or overall betterness ordering is assigned to each profile of real-valued utility functions. In the qualitative framework developed here, numerical utilities are replaced by the properties they are supposed to represent. This makes it possible to characterize the measurability and interpersonal comparability (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  46.  20
    Normal numbers and completeness results for difference sets.Konstantinos A. Beros - 2017 - Journal of Symbolic Logic 82 (1):247-257.
    We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes${\cal D}_2 \left( {{\bf{\Pi }}_3^0 } \right)$and${\cal D}_\omega \left( {{\bf{\Pi }}_3^0 } \right)$, that is, the class of sets which are 2-differences (respectively,ω-differences) of${\bf{\Pi }}_3^0 $sets.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  47.  41
    Different times: Kant and Brouwer on real numbers.Mark van Atten - unknown
  48.  54
    What Numbers Are Real?Kenneth L. Manders - 1986 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:253 - 269.
    We suggest that there can be epistemologically significant reasons why certain mathematical structures - such as the Real numbers - are more important than others. We explore several contexts in which considerations bearing on the choice of a fundamental numerical domain might arise. 1) Set theory. 2) Historical cases of extension of mathematical domains - why were negative numbers resisted, and why should we accept them as part of our fundamental numerical domain? 3) Using fewer reals in (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  49. Science, Pseudoscience, and Science Falsely So-CaIIed.Daniel P. Thurs & Ronald L. Numbers - 2013 - In Massimo Pigliucci & Maarten Boudry (eds.), Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. University of Chicago Press. pp. 121.
    This chapter presents a historical analysis of pseudoscience, tracking down the coinage and currency of the term and explaining its shifting meaning in tandem with the emerging historical identity of science. The discussions cover the invention of pseudoscience; science and pseudoscience in the late nineteenth century; pseudoscience in the new century; and pseudoscience and its critics in the late twentieth century.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  50. Edited volumes-the scientific enterprise in America.Ronald L. Numbers & Charles E. Rosenberg - 1998 - History and Philosophy of the Life Sciences 20 (3):382-384.
     
    Export citation  
     
    Bookmark  
1 — 50 / 943