Ontology of Sets

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk, University of Calgary)
Related categories

41 found
Order:
  1. Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  2. Classes Are States of Affairs.D. M. Armstrong - 1991 - Mind 100 (2):189-200.
  3. Theory of Multiplicities.F. G. Asenjo - 1965 - Logique Et Analyse 8:105-110.
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography   3 citations  
  4. Natura czasoprzestrzeni a istnienie zbiorów.Zdzisław Augustynek - 1995 - Filozofia Nauki 1.
    This paper tries to prove two statements. Firstly, that set-theoretic positions in the controversy on the ontic nature of space-time logically imply set-theoretic realism. Secondly, thatmereological positions in this controversy give set-theoretic nominalism an appearance of verisimilitude.
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography  
  5. Must We Believe in Set Theory?George Boolos - 1998 - In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. pp. 120-132.
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography   5 citations  
  6. Reply to Charles Parsons' ``Sets and Classes''.George Boolos - 1998 - In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. pp. 30-36.
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography   5 citations  
  7. Universality in Set Theories.Manuel Bremer - 2010 - Ontos.
    The book discusses the fate of universality and a universal set in several set theories.
    Remove from this list  
    Translate
      Direct download  
     
    Export citation  
     
    My bibliography  
  8. Set Theory, Type Theory, and Absolute Generality.Salvatore Florio & Stewart Shapiro - 2014 - Mind 123 (489):157-174.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or that neither (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  9. Structuralism and Its Ontology.Marc Gasser - 2015 - Ergo, an Open Access Journal of Philosophy 2 (1):1-26.
    A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of view: its proponents (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  10. A Note on Gabriel Uzquiano's 'Varieties of Indefinite Extensibility'.S. Hewitt - unknown
    It is argued that Gabriel Uzquiano's approach to set-theoretic indefinite extensibility is a version of in rebus structuralism, and therefore suffers from a vacuity problem.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  11. When Do Some Things Form a Set?Simon Hewitt - 2015 - Philosophia Mathematica 23 (3):311-337.
    This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    My bibliography  
  12. Where Do Sets Come From?Harold T. Hodes - 1991 - Journal of Symbolic Logic 56 (1):150-175.
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  13. The Reception of Russell’s Paradox in Early Phenomenology and the School of Brentano: The Case of Husserl’s Manuscript A I 35α.Carlo Ierna - 2016 - In Guillermo E. Rosado Haddock (ed.), Husserl and Analytic Philosophy. De Gruyter. pp. 119-142.
  14. Mathematical Instrumentalism Meets the Conjunction Objection.James Hawthorne - 1996 - Journal of Philosophical Logic 25 (4):363-397.
    Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instnrunentalism escapes the conjunction objection unscathed.
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    My bibliography  
  15. Logic, Logic, and Logic.Richard Jeffrey (ed.) - 1998 - Harvard University Press.
  16. Nine Kinds of Number.John-Michael Kuczynski - 2016 - JOHN-MICHAEL KUCZYNSKI.
    There are nine kinds of number: cardinal (measure of class size), ordinal (corresponds to position), generalized ordinal (position in multidimensional discrete manifold), signed (relation between cardinals), rational (different kind of relation between cardinals), real (limit), complex (pair of reals), transfinite (size of reflexive class), and dimension (measure of complexity.
    Remove from this list  
    Translate
      Direct download  
     
    Export citation  
     
    My bibliography  
  17. Carnap, Quine, Quantification and Ontology.Gregory Lavers - 2015 - In Alessandro Torza (ed.), Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language. Springer.
    Abstract At the time of The Logical Syntax of Language (Syntax), Quine was, in his own words, a disciple of Carnap’s who read this work page by page as it issued from Ina Carnap’s typewriter. The present paper will show that there were serious problems with how Syntax dealt with ontological claims. These problems were especially pronounced when Carnap attempted to deal with higher order quantification. Carnap, at the time, viewed all talk of reference as being part of the misleading (...)
    Remove from this list  
     
    Export citation  
     
    My bibliography  
  18. Accommodating the Informal Notion of Class Within the Framework of Lesniewski's Ontology.Czesław Lejewski - 1985 - Dialectica 39 (3):217-241.
    SummaryInterpreted distributively the sentence‘Indiana is a member of the class of American federal states’means the same as‘Indiana is an American federal state’. In accordance with the collective sense of class expressions the sentence can be understood as implying that Indiana is a part of the country whose capital city is Washington. Neither interpretation appears to accommodate all the intuitions connected with the informal notion of class. A closer accommodation can be achieved, it seems, if class expressions are interpreted as verb‐like (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  19. Conjoining Mathematical Empiricism with Mathematical Realism: Maddy's Account of Set Perception Revisited.Alex Levine - 2005 - Synthese 145 (3):425 - 448.
    Penelope Maddy’s original solution to the dilemma posed by Benacerraf in his (1973) ‘Mathematical Truth’ was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy’s (1990) account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    My bibliography  
  20. Parts of Classes.David Lewis - 1991 - Blackwell.
  21. Category Theory as an Autonomous Foundation.Øystein Linnebo & Richard Pettigrew - 2011 - Philosophia Mathematica 19 (3):227-254.
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other (...)
    Remove from this list   Direct download (13 more)  
     
    Export citation  
     
    My bibliography   5 citations  
  22. Husserl’s Manuscript A I 35.Dieter Lohmar & Carlo Ierna - 2016 - In Guillermo E. Rosado Haddock (ed.), Husserl and Analytic Philosophy. De Gruyter. pp. 289-320.
  23. To Be or to Be Not, That is the Dilemma.Juan José Luetich - 2012 - Identification Transactions of The Luventicus Academy (ISSN 1666-7581) 1 (1):4.
    A set is precisely defined. A given element either belongs or not to a set. However, since all of the elements being considered belong to the universe, if the element does not belong to the set, it belongs to its complement, that is, what remains after all of the elements from the set are removed from the universe.
    Remove from this list  
    Translate
      Direct download  
     
    Export citation  
     
    My bibliography  
  24. WHAT CAN A CATEGORICITY THEOREM TELL US?T. Meadows - 2013 - Review of Symbolic Logic (3):524-544.
    f The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  25. The Argument From Collections.Christopher Menzel - forthcoming - In J. Walls & T. Dougherty (eds.), Two Dozen (or so) Arguments for God: The Plantinga Project. Oxford University Press.
    Very broadly, an argument from collections is an argument that purports to show that our beliefs about sets imply — in some sense — the existence of God. Plantinga (2007) first sketched such an argument in “Two Dozen” and filled it out somewhat in his 2011 monograph Where the Conflict Really Lies: Religion, Science, and Naturalism. In this paper I reconstruct what strikes me as the most plausible version of Plantinga’s argument. While it is a good argument in at least (...)
    Remove from this list   Direct download  
     
    Export citation  
     
    My bibliography  
  26. Wide Sets, ZFCU, and the Iterative Conception.Christopher Menzel - 2014 - Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  27. Cantor on Infinity in Nature, Number, and the Divine Mind.Anne Newstead - 2009 - American Catholic Philosophical Quarterly 83 (4):533-553.
    The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    My bibliography  
  28. Intertwining Metaphysics and Mathematics: The Development of Georg Cantor's Set Theory 1871-1887.Anne Newstead - 2008 - Review of Contemporary Philosophy 7:35-55.
  29. Aristotle and Modern Mathematical Theories of the Continuum.Anne Newstead - 2001 - In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
    Remove from this list  
    Translate
      Direct download  
     
    Export citation  
     
    My bibliography  
  30. Actual Versus Potential Infinity (BPhil Manuscript.).Anne Newstead - 1997 - Dissertation, University of Oxford
    Does mathematical practice require the existence of actual infinities, or are potential infinities enough? Contrasting points of view are examined in depth, concentrating on Aristotle’s arguments against actual infinities, Cantor’s attempts to refute Aristotle, and concluding with Zermelo’s assertion of the primacy of potential infinity in mathematics.
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography  
  31. Motivating Reductionism About Sets.Alexander Paseau - 2008 - Australasian Journal of Philosophy 86 (2):295 – 307.
    The paper raises some difficulties for the typical motivations behind set reductionism, the view that sets are reducible to entities identified independently of set theory.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  32. A Leśniewskian Re-Examination of Goodman's Nominalistic Rejection of Classes.Judith M. Prakel - 1983 - Topoi 2 (1):87-98.
    Remove from this list   Direct download (6 more)  
     
    Export citation  
     
    My bibliography  
  33. Classes as Abstract Entities and the Russell Paradox.Wilfrid Sellars - 1963 - Review of Metaphysics 17 (1):67 - 90.
  34. Some Recent Appeals to Mathematical Experience.Michael J. Shaffer - 2006 - Principia 10 (2):143-170.
    ome recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a number of (...)
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography  
  35. Biometaphysics.Barry Smith - 2009 - In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge.
    While Darwin is commonly supposed to have demonstrated the inapplicability of the Aristotelian ontology of species to biological science, recent developments, especially in the wake of the Human Genome Project, have given rise to a new golden age of classification in which ontological ideas -- as for example in the Gene Ontology, the Cell Ontology, the Protein Ontology, and so forth -- are once again playing an important role. In regard to species, on the other hand, matters are more complex. (...)
    Remove from this list   Direct download  
     
    Export citation  
     
    My bibliography  
  36. Why Numbers Are Sets.Eric Steinhart - 2002 - Synthese 133 (3):343 - 361.
    I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    My bibliography   16 citations  
  37. Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory.Valérie Lynn Therrien - 2012 - Ithaque 10:43-65.
    In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  38. Neologicist Nominalism.Rafal Urbaniak - 2010 - Studia Logica 96 (2):149-173.
    The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...)
    Remove from this list  
    Translate
      Direct download (6 more)  
     
    Export citation  
     
    My bibliography  
  39. Lesniewski's Systems of Logic and Mereology; History and Re-Evaluation.Rafal Urbaniak - 2008 - Dissertation, University of Calgary
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography  
  40. Quantifier Variance and Indefinite Extensibility.Jared Warren - 2017 - Philosophical Review 126 (1):81-122.
    This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions about (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  41. Redukcje ontologiczne w matematyce. Część I.Krzysztof Wójtowicz - 2008 - Filozofia Nauki 3.
    The article is the first part of a series of papers devoted to the problem of ontological reductions in mathematics – in particular, of choosing the basic category of mathematical entities. The received view is that such a category is provided by set theory, which serves as the ontological framework for the whole of mathematics (as all mathematical entities can be represented as sets). However, from the point of view of "naive mathematical realism" we should rather think of the mathematical (...)
    Remove from this list  
    Translate
     
     
    Export citation  
     
    My bibliography