•  4
    Infinite Wordle and the mastermind numbers
    Mathematical Logic Quarterly. forthcoming.
    I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game‐theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of n letters, in…Read more
  •  14
    After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal κ is supercompact if and only if every …Read more
  •  17
    An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical cont…Read more
  •  3
    Proof and the art of mathematics
    The MIT Press. 2020.
    A textbook for students who are learning how to write a mathematical proof, a validation of the truth of a mathematical statement.
  •  15
    An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs. This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory,…Read more
  •  27
    Choiceless large cardinals and set‐theoretic potentialism
    with Raffaella Cutolo
    Mathematical Logic Quarterly 68 (4): 409-415. 2022.
    We define a potentialist system of ‐structures, i.e., a collection of possible worlds in the language of connected by a binary accessibility relation, achieving a potentialist account of the full background set‐theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just. It turns out that the propositional modal assertions which are valid at every world of our syst…Read more
  •  18
    The σ1-definable universal finite sequence
    with Kameryn J. Williams
    Journal of Symbolic Logic 87 (2): 783-801. 2022.
    We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a model in which the sequence is t. O…Read more
  •  22
    Bi-interpretation in weak set theories
    with Alfredo Roque Freire
    Journal of Symbolic Logic 86 (2): 609-634. 2021.
    In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo–Fraenkel set theory $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-foun…Read more
  •  24
    The exact strength of the class forcing theorem
    with Victoria Gitman, Peter Holy, Philipp Schlicht, and Kameryn J. Williams
    Journal of Symbolic Logic 85 (3): 869-905. 2020.
    The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$.…Read more
  •  7
    When does every definable nonempty set have a definable element?
    with François G. Dorais
    Mathematical Logic Quarterly 65 (4): 407-411. 2019.
    The assertion that every definable set has a definable element is equivalent over to the principle, and indeed, we prove, so is the assertion merely that every Π2‐definable set has an ordinal‐definable element. Meanwhile, every model of has a forcing extension satisfying in which every Σ2‐definable set has an ordinal‐definable element. Similar results hold for and and other natural instances of.
  •  153
    We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-…Read more
  •  15
    The implicitly constructible universe
    with Marcia J. Groszek
    Journal of Symbolic Logic 84 (4): 1403-1421. 2019.
    We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe Imp, which they introduced in [5]. Specifically, we show that it is relatively consistent with ZFC that $$Imp = \neg {\rm{CH}}$$, that $Imp \ne {\rm{HOD}}$, and that $$Imp \models V \ne Imp$$, or in other words, that $\left^{Imp} \ne Imp$.
  •  289
    Inner-Model Reflection Principles
    with Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Jonas Reitz, and Ralf Schindler
    Studia Logica 108 (3): 573-595. 2020.
    We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width re…Read more
  •  22
    Set-theoretic blockchains
    with Miha E. Habič, Lukas Daniel Klausner, Jonathan Verner, and Kameryn J. Williams
    Archive for Mathematical Logic 58 (7-8): 965-997. 2019.
    Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite po…Read more
  •  25
    Ehrenfeucht’s Lemma in Set Theory
    with Gunter Fuchs and Victoria Gitman
    Notre Dame Journal of Formal Logic 59 (3): 355-370. 2018.
    Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by adding a C…Read more
  •  121
    A Natural Model of the Multiverse Axioms
    Notre Dame Journal of Formal Logic 51 (4): 475-484. 2010.
    If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins
  •  83
    Inner models with large cardinal features usually obtained by forcing
    with Arthur W. Apter and Victoria Gitman
    Archive for Mathematical Logic 51 (3-4): 257-283. 2012.
    We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compac…Read more
  •  16
    A model of the generic Vopěnka principle in which the ordinals are not Mahlo
    Archive for Mathematical Logic 58 (1-2): 245-265. 2019.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, G…Read more
  •  36
    Set-theoretic mereology
    with Makoto Kikuchi
    Logic and Logical Philosophy 25 (3): 285-308. 2016.
    We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of s…Read more
  •  13
    Strongly uplifting cardinals and the boldface resurrection axioms
    with Thomas A. Johnstone
    Archive for Mathematical Logic 56 (7-8): 1115-1133. 2017.
    We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
  •  26
    The Set-theoretic Multiverse : A Natural Context for Set Theory
    Annals of the Japan Association for Philosophy of Science 19 37-55. 2011.
  •  9
    Incomparable ω 1 ‐like models of set theory
    with Gunter Fuchs and Victoria Gitman
    Mathematical Logic Quarterly 63 (1-2): 66-76. 2017.
    We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω1‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω1‐like models of, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω1‐like model of that does not embed into its own constructible unive…Read more
  •  89
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
  •  47
    Small forcing makes any cardinal superdestructible
    Journal of Symbolic Logic 63 (1): 51-58. 1998.
    Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal κ becomes superdestructible--any further
  •  68
    New inconsistencies in infinite utilitarianism: Is every world good, bad or neutral?
    with Donniell Fishkind and Barbara Montero
    Australasian Journal of Philosophy 80 (2). 2002.
    In the context of worlds with infinitely many bearers of utility, we argue that several collections of natural Utilitarian principles--principles which are certainly true in the classical finite Utilitarian context and which any Utilitarian would find appealing--are inconsistent.
  • Pf= NPf almost everywhere
    with P. D. Welch
    Mathematical Logic Quarterly 49 (5): 536-540. 2003.
  •  59
    Exactly controlling the non-supercompact strongly compact cardinals
    with Arthur W. Apter
    Journal of Symbolic Logic 68 (2): 669-688. 2003.
    We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set o…Read more