26 found
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  1.  16
    Representation of Functions and Total Antisymmetric Relations in Monadic Third Order Logic.M. Randall Holmes - 2019 - Journal of Philosophical Logic 48 (2):263-278.
    We analyze the representation of binary relations in general, and in particular of functions and of total antisymmetric relations, in monadic third order logic, that is, the simple typed theory of sets with three types. We show that there is no general representation of functions or of total antisymmetric relations in this theory. We present partial representations of functions and of total antisymmetric relations which work for large classes of these relations, and show that there is an adequate representation of (...)
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  2.  56
    Strong axioms of infinity in NFU.M. Randall Holmes - 2001 - Journal of Symbolic Logic 66 (1):87-116.
    This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency and (...)
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  3.  39
    The equivalence of NF-Style set theories with "tangled" theories; the construction of ω-models of predicative NF (and more).M. Randall Holmes - 1995 - Journal of Symbolic Logic 60 (1):178-190.
    An ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory "New Foundations" (NF) is constructed. Marcel Crabbe has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for (...)
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  4.  88
    Alternative axiomatic set theories.M. Randall Holmes - 2008 - Stanford Encyclopedia of Philosophy.
  5.  9
    Strong axioms of infinity in NFU.M. Randall Holmes - 2001 - Journal of Symbolic Logic 66 (1):87-116.
    This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency and (...)
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  6.  28
    Systems of combinatory logic related to Quine's ‘New Foundations’.M. Randall Holmes - 1991 - Annals of Pure and Applied Logic 53 (2):103-133.
    Systems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are not constructive.
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  7.  26
    The Philosophy of Set Theory, an Historical Introduction to Cantor's Paradise.M. Randall Holmes - 2006 - Bulletin of Symbolic Logic 12 (4):601-604.
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  8.  43
    The Usual Model Construction for NFU Preserves Information.M. Randall Holmes - 2012 - Notre Dame Journal of Formal Logic 53 (4):571-580.
    The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo–Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper is that the restriction (...)
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  9.  22
    Subsystems of Quine's "New Foundations" with Predicativity Restrictions.M. Randall Holmes - 1999 - Notre Dame Journal of Formal Logic 40 (2):183-196.
    This paper presents an exposition of subsystems and of Quine's , originally defined and shown to be consistent by Crabbé, along with related systems and of type theory. A proof that (and so ) interpret the ramified theory of types is presented (this is a simplified exposition of a result of Crabbé). The new result that the consistency strength of is the same as that of is demonstrated. It will also be shown that cannot be finitely axiomatized (as can and (...)
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  10.  30
    The Axiom Scheme of Acyclic Comprehension.Zuhair Al-Johar, M. Randall Holmes & Nathan Bowler - 2014 - Notre Dame Journal of Formal Logic 55 (1):11-24.
  11.  26
    On hereditarily small sets in ZF.M. Randall Holmes - 2014 - Mathematical Logic Quarterly 60 (3):228-229.
    We show in (the usual set theory without Choice) that for any set X, the collection of sets Y such that each element of the transitive closure of is strictly smaller in size than X (the collection of sets hereditarily smaller than X) is a set. This result has been shown by Jech in the case (where the collection under consideration is the set of hereditarily countable sets).
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  12.  25
    Systems of combinatory logic related to predicative and ‘mildly impredicative’ fragments of Quine's ‘New Foundations’.M. Randall Holmes - 1993 - Annals of Pure and Applied Logic 59 (1):45-53.
    This paper extends the results of an earlier paper by the author . New subsystems of the combinatory logic TRC shown in that paper to be equivalent to NF are introduced; these systems are analogous to subsystems of NF with predicativity restrictions on set comprehension introduced and shown to be consistent by Crabbé. For one of these systems, an exact equivalence in consistency strength and expressive power with the analogous subsystem of NF is established.
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  13.  46
    Paradoxes in double extension set theories.M. Randall Holmes - 2004 - Studia Logica 77 (1):41 - 57.
    Three systems of double extension set theory have been proposed by Andrzej Kisielewicz in two papers. In this paper, it is shown that the two stronger systems are inconsistent, and that the third, weakest system does not admit extensionality for general sets or the use of general sets as parameters in its comprehension scheme. The parameter-free version of the comprehension principle of double extension set theory is also shown to be inconsistent with extensionality. The definitions of the systems and a (...)
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  14.  47
    Automated type-checking for the ramified theory of types of the Principia Mathematica of Russell and Whitehead.M. Randall Holmes - unknown
    This paper described a formal theory of type judgments for propositional logic notations of PM; I felt the need of my own automated type checker to check their examples. The type checker I wrote did indeed serve to help me referee the paper, but also took a rather different approach to notation and typing for propositional functions of PM, which proved worth writing up independently in our own paper: Holmes, M. Randall, “Polymorphic type– checking for the ramified theory of types (...)
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  15.  75
    Errata in "strong axioms of infinity in NFU".M. Randall Holmes - 2001 - Journal of Symbolic Logic 66 (4):1974.
    Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1 , 87--116. Project Euclid: euclid.jsl/1183746361.
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  16.  60
    Foundations of mathematics in polymorphic type theory.M. Randall Holmes - 2001 - Topoi 20 (1):29-52.
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  17.  48
    Karel Lambert, free logic: Selected essays.M. Randall Holmes - 2003 - Studia Logica 75 (3):413-419.
  18.  52
    Polymorphic type checking for the type theory of the Principia Mathematica of Russell and Whitehead.M. Randall Holmes - unknown
    This is a brief report on results reported at length in our paper [2], made for the purpose of a presentation at the workshop to be held in November 2011 in Cambridge on the Principia Mathematica of Russell and Whitehead ([?], hereinafter referred to briefly as PM ). That paper grew out of a reading of the paper [3] of Kamareddine, Nederpelt, and Laan. We refereed this paper and found it useful for checking their examples to write our own independent (...)
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  19.  40
    Symmetry as a Criterion for Comprehension Motivating Quine’s ‘New Foundations’.M. Randall Holmes - 2008 - Studia Logica 88 (2):195 - 213.
    A common objection to Quine’s set theory “New Foundations” is that it is inadequately motivated because the restriction on comprehension which appears to avert paradox is a syntactical trick. We present a semantic criterion for determining whether a class is a set (a kind of symmetry) which motivates NF.
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  20.  21
    Symmetry as a Criterion for Comprehension Motivating Quine’s ‘New Foundations’.M. Randall Holmes - 2008 - Studia Logica 88 (2):195-213.
    A common objection to Quine's set theory "New Foundations" is that it is inadequately motivated because the restriction on comprehension which appears to avert paradox is a syntactical trick. We present a semantic criterion for determining whether a class is a set which motivates NF.
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  21.  36
    The Structure of the Ordinals and the Interpretation of ZF in Double Extension Set Theory.M. Randall Holmes - 2005 - Studia Logica 79 (3):357-372.
    Andrzej Kisielewicz has proposed three systems of double extension set theory of which we have shown two to be inconsistent in an earlier paper. Kisielewicz presented an argument that the remaining system interprets ZF, which is defective: it actually shows that the surviving possibly consistent system of double extension set theory interprets ZF with Separation and Comprehension restricted to 0 formulas. We show that this system does interpret ZF, using an analysis of the structure of the ordinals.
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  22.  13
    Karel Lambert, Free Logic: Selected Essays. [REVIEW]M. Randall Holmes - 2003 - Studia Logica 75 (3):413-419.
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  23.  36
    T. E. Forster. Set theory with a universal set. Exploring an untyped universe. Second edition of LVIII 725. Oxford logic guides, no. 31. Clarendon Press, Oxford University Press, Oxford and New York1995, x + 166 pp. [REVIEW]M. Randall Holmes - 1996 - Journal of Symbolic Logic 61 (4):1392-1393.
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  24.  23
    Mary Tiles. The philosophy of set theory, an historical introduction to Cantor's paradise. Courier Dover Publications, 2004, xiii + 239 pp. [REVIEW]M. Randall Holmes - 2006 - Bulletin of Symbolic Logic 12 (4):601-604.
  25.  17
    Review of Manuel Bremer, Universality in Set Theories: A Study in Formal Ontology[REVIEW]M. Randall Holmes - 2010 - Notre Dame Philosophical Reviews 2010 (8).
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  26. Review: T. E. Forster, Set Theory with a Universal Set. Exploring an Untyped Universe. [REVIEW]M. Randall Holmes - 1996 - Journal of Symbolic Logic 61 (4):1392-1393.