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  1. Zermelian Extensibility.Andrew Bacon - manuscript
    According to an influential idea in the philosophy of set theory, certain mathematical concepts, such as the notion of a well-order and set, are indefinitely extensible. Following Parsons (1983), this has often been cashed out in modal terms. This paper explores instead an extensional articulation of the idea, formulated in higher-order logic, that flat-footedly formalizes some remarks of Zermelo. The resulting picture is incompatible with the idea that the entire universe can be well-ordered, but entirely consistent with the idea that (...)
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  2. A Reassessment of Cantorian Abstraction based on the ε-operator.Nicola Bonatti - forthcoming - Synthese.
    Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor's proposal based upon the set theoretic framework of Bourbaki - called BK - which is a First-order set theory extended with Hilbert's ε-operator. Moreover, it is argued that the BK system and the ε-operator provide a faithful reconstruction of Cantor's insights on cardinal numbers. (...)
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  3. Against Cumulative Type Theory.Tim Button & Robert Trueman - 2022 - Review of Symbolic Logic 15 (4):907-49.
    Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions imposed by (...)
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  4. On Number-Set Identity: A Study.Sean C. Ebels-Duggan - 2022 - Philosophia Mathematica 30 (2):223-244.
    Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.
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  5. TRES TEOREMAS SOBRE CARDINALES MEDIBLES.Franklin Galindo - 2021 - Mixba'al. Revista Metropolitana de Matemáticas 12 (1):15-31.
    El estudio de los "cardinales grandes" es uno de los principales temas de investigación de la teoría de conjuntos y de la teoría de modelos que ha contribuido con el desarrollo de dichas disciplinas. Existe una gran variedad de tales cardinales, por ejemplo cardinales inaccesibles, débilmente compactos, Ramsey, medibles, supercompactos, etc. Tres valiosos teoremas clásicos sobre cardinales medibles son los siguientes: (i) compacidad débil, (ii) Si κ es un cardinal medible, entonces κ es un cardinal inaccesible y existen κ cardinales (...)
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  6. Measuring the intelligence of an idealized mechanical knowing agent.Samuel Alexander - 2020 - Lecture Notes in Computer Science 12226.
    We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define real-number intelligence levels of compli- cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows to be codes (...)
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  7. AGI and the Knight-Darwin Law: why idealized AGI reproduction requires collaboration.Samuel Alexander - 2020 - Agi.
    Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and (...)
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  8. The logic of comparative cardinality.Yifeng Ding, Matthew Harrison-Trainor & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (3):972-1005.
    This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.
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  9. Neo-Fregeanism and the Burali-Forti Paradox.Ian Rumfitt - 2018 - In Ivette Fred Rivera & Jessica Leech (eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale. Oxford, England: Oxford University Press. pp. 188-223.
    Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a predicative logic. (...)
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  10. Size and Function.Bruno Whittle - 2018 - Erkenntnis 83 (4):853-873.
    Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for (...)
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  11. Gödel's Argument for Cantorian Cardinality.Matthew W. Parker - 2017 - Noûs 53 (2):375-393.
    On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's (...)
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  12. Basic Principles of Mathematical Logic.John-Michael Kuczynski - 2016 - Amazon Digital Services LLC.
    This book concisely states the main laws and precepts of formal logic along with their immediate corollaries. Commentary is kept to a minimum.
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  13. Mathematical Logic and Formal Arithmetic: Key Definitions and Principles.John-Michael Kuczynski - 2016 - Amazon Digital Services LLC.
    This books states, as clearly and concisely as possible, the most fundamental principles of set-theory and mathematical logic. Included is an original proof of the incompleteness of formal logic. Also included are clear and rigorous definitions of the primary arithmetical operations, as well as clear expositions of the arithmetic of transfinite cardinals.
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  14. Reply to Vann McGee's 'Whittle's Assault on Cantor’s Paradise'.Bruno Whittle - 2015 - In Oxford Studies in Metaphysics: Volume 9. Oxford University Press. pp. 33-41.
    This is a reply to Vann McGee’s response to my paper, ‘On Infinite Size’.
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  15. (1 other version)On Infinite Size.Bruno Whittle - 2015 - In Oxford Studies in Metaphysics: Volume 9. Oxford University Press. pp. 3-19.
    Cantor showed that there are infinite sets that do not have one-to-one correspondences between them. The standard understanding of this result is that it shows that there are different sizes of infinity. This paper challenges this standard understanding, and argues, more generally, that we do not have any reason to think that there are different sizes of infinity. Two arguments are given against the claim that Cantor established that there are different such sizes: one involves an analogy between Cantor’s result (...)
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  16. (1 other version)On infinite size.Bruno Whittle - 2015 - Oxford Studies in Metaphysics 9:3-19.
    This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The chapter challenges this, arguing (...)
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  17. Eliminating the ordinals from proofs. An analysis of transfinite recursion.Edoardo Rivello - 2014 - In Proceedings of the Conference "Philosophy, Mathematics, Linguistics. Aspects of Interaction", St. Petersburg, April 21-25, 2014. pp. 174-184.
    Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination which characterises (...)
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  18. Finite Cardinals in Quasi-set Theory.Jonas R. Becker Arenhart - 2012 - Studia Logica 100 (3):437-452.
    Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to them. According to (...)
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  19. Cantor’s Proof in the Full Definable Universe.Laureano Luna & William Taylor - 2010 - Australasian Journal of Logic 9:10-25.
    Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...)
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  20. Set Theory and its Philosophy: A Critical Introduction.Michael D. Potter - 2004 - Oxford, England: Oxford University Press.
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...)
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  21. Inverse Operations with Transfinite Numbers and the Kalam Cosmological Argument.Graham Oppy - 1995 - International Philosophical Quarterly 35 (2):219-221.
    William Lane Craig has argued that there cannot be actual infinities because inverse operations are not well-defined for infinities. I point out that, in fact, there are mathematical systems in which inverse operations for infinities are well-defined. In particular, the theory introduced in John Conway's *On Numbers and Games* yields a well-defined field that includes all of Cantor's transfinite numbers.
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  22. Reply to an Objection to Animal Rights.Joseph S. Fulda - 1992 - Journal of Value Inquiry 26 (1):87-88.
    Notwithstanding the numerous errors in this piece, the core teaching remains unscathed: Arithmetic (or any other branch of mathematics) cannot do moral work. If it appears otherwise, that simply means some nonstandard version of the relevant area of mathematics will work. -/- Negative results can indeed sometimes be shown using mathematics, but not on such fundamental normative questions as whether something/someone has rights. Also, mathematics can put into relief, sometimes, a fundamental normative question, even though it cannot resolve it.
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  23. Cantor and the Burali-Forti Paradox.Christopher Menzel - 1984 - The Monist 67 (1):92-107.
    In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind. This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundations of set theory to be shaken, he showed no apparent concern over the paradox and its implications (...)
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