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# Axioms of Set Theory, Misc

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1. Is (un)countabilism restrictive?Neil Barton - manuscript
Let's suppose you think that there are no uncountable sets. Have you adopted a restrictive position? It is certainly tempting to say yes---you've prohibited the existence of certain kinds of large set. This paper argues that this intuition can be challenged. Instead, I argue that there are some considerations based on a formal notion of restrictiveness which suggest that it is restrictive to hold that there are uncountable sets.

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2. Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.
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3. Wand/Set Theories: A realization of Conway's mathematicians' liberation movement, with an application to Church's set theory with a universal set.Tim Button - forthcoming - Journal of Symbolic Logic:1-46.
Here is a template for introducing mathematical objects: “Objects are found in stages. For every stage S: (1) for any things found before S, you find at S the bland set whose members are exactly those things; (2) for anything, x, which was found before S, you find at S the result of tapping x with any magic wand (provided that the result is not itself a bland set); you find nothing else at S.” -/- This Template has rich applications, (...)
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4. Observation and Intuition.Justin Clarke-Doane & Avner Ash - forthcoming - In Carolin Antos, Neil Barton & Venturi Giorgio (eds.), Palgrave Companion to the Philosophy of Set Theory.
The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...)
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5. The iterative conception of function and the iterative conception of set.Tim Button - 2023 - In Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory. Palgrave.
Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we (...)

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6. A Logical Foundation for Potentialist Set Theory.Sharon Berry - 2022 - Cambridge University Press.
In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further defends (...)

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7. Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets.Tim Button - 2021 - Bulletin of Symbolic Logic 27 (4):436-460.
The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories (...)
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8. Finitist set theory in ontological modeling.Avril Styrman & Aapo Halko - 2018 - Applied ontology 13 (2):107-133.
This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as (...)
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9. Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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10. Russell’s method of analysis and the axioms of mathematics.Lydia Patton - 2017 - In Sandra Lapointe Christopher Pincock (ed.), Innovations in the History of Analytical Philosophy. London: Palgrave-Macmillan. pp. 105-126.
In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...)
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11. Why is the universe of sets not a set?Zeynep Soysal - 2017 - Synthese 197 (2):575-597.
According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I (...)
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12. Ineffability within the limits of abstraction alone.Stewart Shapiro & Gabriel Uzquiano - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics. Oxford University Press.
The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that (...)
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13. Objectivity in Ethics and Mathematics.Justin Clarke-Doane - 2015 - Proceedings of the Aristotelian Society: The Virtual Issue 3.
How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.

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14. Maddy, Penelope, Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford: Oxford University Press, 2011, pp. x + 150, £29/us\$45 (hardback). [REVIEW]Jeffrey W. Roland - 2013 - Australasian Journal of Philosophy 91 (4):809-812.
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15. What is Absolute Undecidability?†.Justin Clarke-Doane - 2012 - Noûs 47 (3):467-481.
It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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16. Fraenkel's axiom of restriction: Axiom choice, intended models and categoricity.Georg Schiemer - 2010 - In Benedikt L.öwe & Thomas Müller (eds.), PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications. pp. 307{340.

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17. Too naturalist and not naturalist enough: Reply to Horsten.Luca Incurvati - 2008 - Erkenntnis 69 (2):261 - 274.
Leon Horsten has recently claimed that the class of mathematical truths coincides with the class of theorems of ZFC. I argue that the naturalistic character of Horsten’s proposal undermines his contention that this claim constitutes an analogue of a thesis that Daniel Isaacson has advanced for PA. I argue, moreover, that Horsten’s defence of his claim against an obvious objection makes use of a distinction which is not available to him given his naturalistic approach. I suggest a way out of (...)
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18. Frege Meets Zermelo: A Perspective on Ineffability and Reflection.Stewart Shapiro - 2008 - Review of Symbolic Logic 1 (2):241-266.
1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.
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19. Boolos on the justification of set theory.Alexander Paseau - 2007 - Philosophia Mathematica 15 (1):30-53.
George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.
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20. Models of second-order zermelo set theory.Gabriel Uzquiano - 1999 - Bulletin of Symbolic Logic 5 (3):289-302.
In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of (...)
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21. What is Cantor's Continuum Problem?Kurt Gödel - 1983 - In Paul Benacerraf & Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings (2nd Edition). Cambridge: Cambridge University Press. pp. 470-485.
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22. A reduction of the axioms for the theory of propositional types.Peter Andrews - 1963 - Fundamenta Mathematicae 52:345-350.
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23. Die Axiome der Arithmetik mit besonderer Berücksichtigung der Beziehungen zur Mengenlehre.Kurt Grelling - 1910 - Dissertation, Georg-Augusts-Universität Göttingen