Results for 'Goedel numbers'

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  1. 3. The Monotone Series and Multiplier and Divisor Relative Numbers.Divisor Relative Numbers - 1987 - International Logic Review: Rassegna Internazionale di Logica 15 (1):26.
     
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  2.  60
    A Conversation About Numbers.Charles Sayward - 2002 - Philosophia 29 (1-4):191-209.
    This is a dialogue in which five characters are involved. Various issues in the philosophy of mathematics are discussed. Among those issues are these: numbers as abstract objects, our knowledge of numbers as abstract objects, a proof as showing a mathematical statement to be true as opposed to the statement being true in virtue of having a proof.
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  3. Reference to Numbers in Natural Language.Friederike Moltmann - 2013 - Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are (...)
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  4. Of Numbers and Electrons.Cian Dorr - 2010 - Proceedings of the Aristotelian Society 110 (2pt2):133-181.
    According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the (...)
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  5.  16
    Sustainability Ratings and the Disciplinary Power of the Ideology of Numbers.Mohamed Chelli & Yves Gendron - 2013 - Journal of Business Ethics 112 (2):187-203.
    The main purpose of this paper is to better understand how sustainability rating agencies, through discourse, promote an “ideology of numbers” that ultimately aims to establish a regime of normalization governing social and environmental performance. Drawing on Thompson’s (Ideology and modern culture: Critical social theory in the era of mass communication, 1990 ) modes of operation of ideology, we examine the extent to which, and how, the ideology of numbers is reflected on websites and public documents published by (...)
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  6. Taurek, Numbers and Probabilities.Rob Lawlor - 2006 - Ethical Theory and Moral Practice 9 (2):149 - 166.
    In his paper, “Should the Numbers Count?" John Taurek imagines that we are in a position such that we can either save a group of five people, or we can save one individual, David. We cannot save David and the five. This is because they each require a life-saving drug. However, David needs all of the drug if he is to survive, while the other five need only a fifth each.Typically, people have argued as if there was a choice (...)
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  7.  82
    Defusing Easy Arguments for Numbers.Brendan Balcerak Jackson - 2013 - Linguistics and Philosophy 36 (6):447-461.
    Pairs of sentences like the following pose a problem for ontology: (1) Jupiter has four moons. (2) The number of moons of Jupiter is four. (2) is intuitively a trivial paraphrase of (1). And yet while (1) seems ontologically innocent, (2) appears to imply the existence of numbers. Thomas Hofweber proposes that we can resolve the puzzle by recognizing that sentence (2) is syntactically derived from, and has the same meaning as, sentence (1). Despite appearances, the expressions ‘the number (...)
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  8.  31
    The Psychology and Philosophy of Natural Numbers.Oliver R. Marshall - 2017 - Philosophia Mathematica:nkx002.
    ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal (...)
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  9.  7
    More on D-Logics of Subspaces of the Rational Numbers.Guram Bezhanishvili & Joel Lucero-Bryan - 2012 - Notre Dame Journal of Formal Logic 53 (3):319-345.
    We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$ . It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4 , continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely (...)
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  10.  19
    The Numbers Count.Peter A. Graham - 2017 - Thought: A Journal of Philosophy 6 (2):129-134.
    Numbers Skeptics deny that when faced with a choice between saving some innocent people from harm and saving a larger number of different, though equally innocent, people from suffering a similar harm you ought to save the larger number. In this article, I aim to put pressure on Numbers Skepticism.
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  11.  46
    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 (...)
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  12.  29
    Frege, the Complex Numbers, and the Identity of Indiscernibles.Wenzel Christian Helmut - 2010 - Logique Et Analyse 53 (209):51-60.
    There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to (...)
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  13.  65
    Don’T Count on Taurek: Vindicating the Case for the Numbers Counting.Yishai Cohen - 2014 - Res Publica 20 (3):245-261.
    Suppose you can save only one of two groups of people from harm, with one person in one group, and five persons in the other group. Are you obligated to save the greater number? While common sense seems to say ‘yes’, the numbers skeptic says ‘no’. Numbers Skepticism has been partly motivated by the anti-consequentialist thought that the goods, harms and well-being of individual people do not aggregate in any morally significant way. However, even many non-consequentialists think that (...)
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  14.  52
    Platonism by the Numbers.Steven M. Duncan - manuscript
    In this paper, I defend traditional Platonic mathematical realism from its contemporary detractors, arguing that numbers, understood as abstract, non-physical objects of rational intuition, are indispensable for the act of counting.
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  15.  49
    The New Problem of Numbers in Morality.Fiona Woollard - 2014 - Ethical Theory and Moral Practice 17 (4):631-641.
    Discussion of the “problem of numbers” in morality has focused almost exclusively on the moral significance of numbers in whom-to-rescue cases: when you can save either of two groups of people, but not both, does the number of people in each group matter morally? I suggest that insufficient attention has been paid to the moral significance of numbers in other types of case. According to common-sense morality, numbers make a difference in cases, like the famous Trolley (...)
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  16.  6
    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.
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  17.  18
    Non-Constructive Properties of the Real Numbers.J. E. Rubin, K. Keremedis & Paul Howard - 2001 - Mathematical Logic Quarterly 47 (3):423-431.
    We study the relationship between various properties of the real numbers and weak choice principles.
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  18.  26
    Philosophy's Numerical Turn: Why the Pythagoreans' Interest in Numbers is Truly Awesome.Catherine Rowett - 2013 - In Dirk Obbink & David Sider (eds.), Doctrine and Doxography: Studies on Heraclitus and Pythagoras. De Gruyter. pp. 3-32.
    Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers—not just its physics but perhaps morality too. (...)
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  19.  18
    Learning to Represent Exact Numbers.Barbara W. Sarnecka - forthcoming - Synthese:1-18.
    This article focuses on how young children acquire concepts for exact, cardinal numbers. I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey. In this (...)
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  20.  43
    Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW]Wojciech Krysztofiak - 2012 - Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in (...)
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  21.  6
    Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions.Melissa DeWolf, Ji Y. Son, Miriam Bassok & Keith J. Holyoak - 2017 - Cognitive Science 41 (8):2053-2088.
    Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...)
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  22. Numbers as Ontologically Dependent Objects - Hume's Principle Revisited.Robert Schwartzkopff - 2011 - Grazer Philosophische Studien 82 (1):353-373.
    Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.
     
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  23.  20
    Egalitarianism, Numbers and the Dreaded Conclusion.Gabriel Wollner - 2012 - Ethical Perspectives 19 (3):399-416.
    Some contractualist egalitarians try to accommodate a concern for numbers by embracing a pluralist strategy. They incorporate the belief that the number of people affected matters for what distribution one ought to bring about by arguing that their primary contractualist concern for justifiability to each may be outweighed by aggregative considerations. The present contribution offers two arguments against such a pluralist strategy. First, I argue that advo- cates of the pluralist strategy are forced to abandon the rationale behind the (...)
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  24.  12
    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 a number of different (...)
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  25.  29
    A Conversation About Numbers and Knowledge.Charles Sayward - 2002 - American Philosophical Quarterly 39 (3):275-287.
    This is a dialogue in the philosophy of mathematics. The dialogue descends from the confident assertion that there are infinitely many numbers to an unresolved bewilderment about how we can know there are any numbers at all. At every turn the dialogue brings us only to realize more fully how little is clear to us in our thinking about mathematics.
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  26.  9
    On the Ordered Dedekind Real Numbers in Toposes.Marcelo E. Coniglio & Luís A. Sbardellini - 2015 - In Edward H. Haeusler, Wagner Sanz & Bruno Lopes (eds.), Why is this a Proof? Festschrift for Luiz Carlos Pereira. College Publications. pp. 87-105.
    In 1996, W. Veldman and F. Waaldijk present a constructive (intuitionistic) proof for the homogeneity of the ordered structure of the Cauchy real numbers, and so this result holds in any topos with natural number object. However, it is well known that the real numbers objects obtained by the traditional constructions of Cauchy sequences and Dedekind cuts are not necessarily isomorphic in an arbitrary topos with natural numbers object. Consequently, Veldman and Waaldijk's result does not apply to (...)
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  27.  3
    Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions.Melissa DeWolf, Ji Y. Son, Miriam Bassok & Keith J. Holyoak - 2017 - Cognitive Science 41 (8):2053-2088.
    Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...)
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  28.  2
    Measurement, “Scriptural Economies,” and Social Justice: Governing HIV/AIDS Treatments by Numbers in a Fragile State, the Central African Republic.Pierre‐Marie David - 2017 - Developing World Bioethics 17 (1):32-39.
    Fragile states have been raising increasing concern among donors since the mid-2000s. The policies of the Global Fund to fight HIV/AIDS, Malaria, and Tuberculosis have not excluded fragile states, and this source has provided financing for these countries according to standardized procedures. They represent interesting cases for exploring the meaning and role of measurement in a globalized context. Measurement in the field of HIV/AIDS and its treatment has given rise to a private outsourcing of expertise and auditing, thereby creating a (...)
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  29.  1
    Ordinal Numbers in Arithmetic Progression.Frederick Bagemihl & F. Bagemihl - 1992 - Mathematical Logic Quarterly 38 (1):525-528.
    The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers γ r, δ ≠ 0.
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  30. Mathematical Logic with Special Reference to the Natural Numbers.S. W. P. Steen - 1972 - Cambridge University Press.
    This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in the main (...)
     
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  31.  70
    Talking About Numbers: Easy Arguments for Mathematical Realism. [REVIEW]Richard Lawrence - 2017 - History and Philosophy of Logic 38 (4):390-394.
  32.  17
    Speed of Adding and Comparing Numbers.Frank Restle - 1970 - Journal of Experimental Psychology 83 (2p1):274.
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  33.  55
    Distinguishing Non-Standard Natural Numbers in a Set Theory Within Łukasiewicz Logic.Shunsuke Yatabe - 2007 - Archive for Mathematical Logic 46 (3-4):281-287.
    In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.
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  34. The Individuation of the Natural Numbers.Øystein Linnebo - 2009 - In Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave.
    It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal (...)
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  35.  67
    Spatial Localization in Quantum Theory Based on Qr-Numbers.John Corbett & Thomas Durt - 2010 - Foundations of Physics 40 (6):607-628.
    We show how trajectories can be reintroduced in quantum mechanics provided that its spatial continuum is modelled by a variable real number (qr-number) continuum. Such a continuum can be constructed using only standard Hilbert space entities. In this approach, the geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare trajectories in this new description of space-time with (...)
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  36.  17
    On Σ‐Definability Without Equality Over the Real Numbers.Andrei S. Morozov & Margarita V. Korovina - 2008 - Mathematical Logic Quarterly 54 (5):535-544.
    In [5] it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for Σ-definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of Σ-definable sets into new definitions of Σ-definable sets in such (...)
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  37.  3
    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.
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  38.  7
    Value of Standard and Very First Variable in Judgments of Reflectance of Grays with Various Ranges of Available Numbers.E. C. Poulton & D. C. V. Simmonds - 1963 - Journal of Experimental Psychology 65 (3):297.
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  39.  6
    Order‐Free Recursion on the Real Numbers.Vasco Brattka - 1997 - Mathematical Logic Quarterly 43 (2):216-234.
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  40.  3
    Effects of Interval Between Successive Numbers and Pattern in Verbal Learning.Charles P. Thompson - 1965 - Journal of Experimental Psychology 70 (6):626.
  41.  4
    Effect of Pattern in Display by Letters and Numerals Upon Acquisition of Serial Lists of Numbers.Allan L. Fingeret & W. J. Brogden - 1973 - Journal of Experimental Psychology 98 (2):339.
  42.  1
    Free Recall of Numbers with High- and Low-Rated Association Values.Stefan Slak - 1970 - Journal of Experimental Psychology 83 (1p1):184.
  43. The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Iamblichus - 1988 - Phanes Press.
  44. Science Without Numbers.Hartry Field - 1980 - Princeton University Press.
    Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.
     
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  45. The Reality of Numbers: A Physicalist's Philosophy of Mathematics.John Bigelow - 1988 - Oxford University Press.
    Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.
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  46. Belief Dependence: How Do the Numbers Count?Zach Barnett - forthcoming - Philosophical Studies:1-23.
    This paper is about how to aggregate outside opinion. If two experts are on one side of an issue, while three experts are on the other side, what should a non-expert believe? Certainly, the non-expert should take into account more than just the numbers. But which other factors are relevant, and why? According to the view developed here, one important factor is whether the experts should have been expected, in advance, to reach the same conclusion. When the agreement of (...)
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  47. Talking About Nothing: Numbers, Hallucinations, and Fictions.Jody Azzouni - 2010 - Oxford University Press.
    Numbers -- Hallucinations -- Fictions -- Scientific languages, ontology, and truth -- Truth conditions and semantics.
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  48.  68
    Transfinite Numbers in Paraconsistent Set Theory.Zach Weber - 2010 - Review of Symbolic Logic 3 (1):71-92.
    This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will (...)
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  49.  80
    Number Words and Reference to Numbers.Katharina Felka - 2014 - Philosophical Studies 168 (1):261-282.
    A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we (...)
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  50.  44
    The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small.Philip Ehrlich - 2012 - Bulletin of Symbolic Logic 18 (1):1-45.
    In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be (...)
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