Frege’s philosophy of mathematics—Interview with Nathan Ormond, December 2021

I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.

The interview concludes with a public comment and question & answer session.

A deflationary account of Fregean abstraction in Zermelo-Fraenkel ZF set theory, Oxford, November 2021

This will be a talk for the Oxford Seminar in the Philosophy of Mathematics, 1 November, 4:30-6:30 GMT. The talk will be held on Zoom (contact the seminar organizers for the Zoom link).

Abstract. The standard treatment of sets and classes in Zermelo-Fraenkel set theory instantiates in many respects the Fregean foundational distinction between objects and concepts, for in set theory we commonly take the sets as objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is often asserted that there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework I have described, it turns out that Basic Law V does hold, and provably so, along with other various Fregean abstraction principles. These principles are consequences of Zermelo-Fraenkel ZF set theory in the context of all its definable classes. Namely, there is an injective mapping from classes to objects, definable in senses I shall explain, associating every first-order parametrically definable class $F$ with a set object $\varepsilon F$, in such a way that Basic Law V is fulfilled: $$\varepsilon F =\varepsilon G\iff\forall x\ (Fx\leftrightarrow Gx).$$ Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument.

Counting to Infinity and Beyond

Anyone can learn to count in the ordinals, even a child, and so let us learn how to count to $\omega^2$, the first compound limit ordinal.

The large-format poster is available:

Some close-up views:

I would like to thank the many people who had made helpful suggestions concerning my poster, including Andrej Bauer and especially Saul Schleimer, who offered many detailed suggestions.

Infinite sets and Foundations—Interviewed on the Daniel Rubin Show

I was interviewed 26 August 2021 by mathematician Daniel Rubin on his show, and we had a lively, wideranging discussion spanning mathematics, infinity, and the philosophy of mathematics. Please enjoy!

Contents

0:00 Intro

2:11 Joel’s background. Interaction between math and philosophy

9:04 Joel’s work; infinite chess.

14:45 Infinite ordinals

22:27 The Cantor-Bendixson process

29:41 Uncountable ordinals

32:10 First order vs. second order theories

41:16 Non-standard analysis

46:57 The ZFC axioms and well-ordering of the reals

58:11 Showing independence of statements. Models and forcing.

1:04:38 Sets, classes, and categories

1:19:22 Is there one true set theory? Are projective sets Lebesgue measurable?

1:30:20 What does set theory look like if certain axioms are rejected?

1:36:06 How to judge philosophical positions about math

1:42:01 Concrete math where set theory becomes relevant. Tarski-Seidenberg on positive polynomials.

1:48:48 Goodstein sequences and the use of infinite ordinals

1:58:43 The state of set theory today

2:01:41 Joel’s recent books

Go check out the other episodes on Daniel’s channel!

The Tennenbaum phenomenon for computable quotient presentations of models of arithmetic and set theory, Shanghai, August 2021

This will be a talk for the conference Fudan Model Theory and Philosophy of Mathematics, held at Fudan University in Shanghai and online, 21-24 August 2021. My talk will take place on Zoom on 23 August 20:00 Beijing time (1pm BST).

Abstract. Tennenbaum famously proved that there is no computable presentation of a nonstandard model of arithmetic or indeed of any model of set theory. In this talk, I shall discuss the Tennenbaum phenomenon as it arises for computable quotient presentations of models. Quotient presentations offer a philosophically attractive treatment of identity, a realm in which questions of identity are not necessarily computable. Objects in the presentation serve in effect as names for objects in the final quotient structure, names that may represent the same or different items in that structure, but one cannot necessarily tell which. Bakhadyr Khoussainov outlined a sweeping vision for quotient presentations in computable model theory and made several conjectures concerning the Tennenbaum phenomenon. In this talk, I shall discuss joint work with Michał Godziszewski that settles and addresses several of these conjectures.

Plenitudinous Primes!

A proof of the infinitude of primes, in meter.

Let’s prove the primes’ infinitude
they do exist in multitude

of quite astounding magnitude
we count the primes in plenitude!

‘Twas proved by Euclid way back when
in Elements he argued then

He proved it with exactitude
in his Book IX he did conclude

for any given finite list
there will be primes the list has missed

to prove this gem let number N
be when you multiply them then

multiply the list for fun,
multiply, and then add one

If into N we shall divide
the listed primes seen in that guide

remainder one each leaves behind
so none as factors could we find

but every number has some prime factor
with our N take such an actor

since it can’t be on the list
we thus deduce more primes exist

No finite list can be complete
and so the claim is in retreat

By iterating this construction
primes do flow in vast deduction

as many as we’d like to see
so infinite the primes must be

Thus proved the primes’ infinitude
in quite enormous multitude

of arb’trarly great magnitude
we count the primes in plenitude!

‘Twas known by Euclid way back then
he proved it in the Elements when

in modern times we know some more
math’s never done, there’s new folklore:

For Bertrand, Chebyshev had said
about the puzzle in his head

and Erdős said it once again:
there’s always a prime between n and 2n.

We proved the primes’ infinitude
they do exist in multitude

of inconceiv’ble magnitude
we count the primes in plentitude!

The primes are plenitudinous
they’re truly multitudinous!


Musical video production of Plenitudinous Primes by the supremely talented Hannah Hoffman!

Ode to Hippasus

I was so glad to be involved with this project of Hannah Hoffman. She had inquired on Twitter whether mathematicians could provide a proof of the irrationality of root two that rhymes. I set off immediately, of course, to answer the challenge. My wife Barbara Gail Montero and our daughter Hypatia and I spent a day thinking, writing, revising, rewriting, rethinking, rewriting, and eventually we had a set lyrics providing the proof, in rhyme and meter. We had wanted specifically to highlight not only the logic of the proof, but also to tell the fateful story of Hippasus, credited with the discovery.

Hannah proceeded to create the amazing musical version:

The diagonal of a square is incommensurable with its side
an astounding fact the Pythagoreans did hide

but Hippasus rebelled and spoke the truth
making his point with irrefutable proof

it’s absurd to suppose that the root of two
is rational, namely, p over q

square both sides and you will see
that twice q squared is the square of p

since p squared is even, then p is as well
now, if p as 2k you alternately spell

2q squared will to 4k squared equate
revealing, when halved, q’s even fate

thus, root two as fraction, p over q
must have numerator and denomerator with factors of two

to lowest terms, therefore, it can’t be reduced
root two is irrational, Hippasus deduced

as retribution for revealing this irrationality
Hippasus, it is said, was drowned in the sea

but his proof live on for the whole world to admire
a truth of elegance that will ever inspire.

Forcing as a computational process, Kobe Set Theory Workshop, March 2021

This was a talk for the Kobe Set Theory Workshop, held on the occasion of Sakaé Fuchino’s retirement, 9-11 March 2021.

Abstract. I shall discuss senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model of set theory $\langle M,\in^M\rangle$, for example, one may in various senses compute $M$-generic filters $G\subset P\in M$ and the corresponding forcing extensions $M[G]$. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory $M$ that lead by the computational process to non-isomorphic forcing extensions $M[G]\not\cong M[G’]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

This is joint work with Russell Miller and Kameryn Williams.

Forcing as a computational process

  • J. D. Hamkins, R. Miller, and K. J. Williams, “Forcing as a computational process,” Mathematics ArXiv, 2020.
    [Bibtex]
    @ARTICLE{HamkinsMillerWilliams:Forcing-as-a-computational-process,
    author = {Joel David Hamkins and Russell Miller and Kameryn J. Williams},
    title = {Forcing as a computational process},
    journal = {Mathematics ArXiv},
    year = {2020},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    source = {},
    doi = {},
    url = {http://jdh.hamkins.org/forcing-as-a-computational-process},
    eprint = {2007.00418},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Definability and the Math Tea argument: must there be numbers we cannot describe or define? University of Warsaw, 22 January 2021

This will be a talk for a new mathematical logic seminar at the University of Warsaw in the Department of Hhilosophy, entitled Epistemic and Semantic Commitments of Foundational Theories, devoted to formal truth theories and implicit commitments of foundational theories as well as their conceptual surroundings.

My talk will be held 22 January 2021, 8 pm CET (7 pm UK), online via Zoom https://us02web.zoom.us/j/83366049995.

Tran Tuan, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons

Abstract. According to the math tea argument, perhaps heard at a good afternoon tea, there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions. Is it correct? In this talk, I shall discuss the phenomenon of pointwise definable structures in mathematics, structures in which every object has a property that only it exhibits. A mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it exhibit different properties. Is there a Leibnizian structure with no definable elements? We shall discuss many interesting elementary examples, eventually working up to the proof that every countable model of set theory has a pointwise definable extension, in which every mathematical object is definable.

Pointwise definable models of set theory

  • [DOI] J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of Symbolic Logic, vol. 78, iss. 1, p. 139–156, 2013.
    [Bibtex]
    @article {HamkinsLinetskyReitz2013:PointwiseDefinableModelsOfSetTheory,
    AUTHOR = {Hamkins, Joel David and Linetsky, David and Reitz, Jonas},
    TITLE = {Pointwise definable models of set theory},
    JOURNAL = {Journal of Symbolic Logic},
    FJOURNAL = {Journal of Symbolic Logic},
    VOLUME = {78},
    YEAR = {2013},
    NUMBER = {1},
    PAGES = {139--156},
    ISSN = {0022-4812},
    MRCLASS = {03E55},
    MRNUMBER = {3087066},
    MRREVIEWER = {Bernhard A. König},
    DOI = {10.2178/jsl.7801090},
    URL = {http://jdh.hamkins.org/pointwisedefinablemodelsofsettheory/},
    eprint = "1105.4597",
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Can there be natural instances of nonlinearity in the hierarchy of consistency strength? UWM Logic Seminar, January 2021

This is a talk for the University of Wisconsin, Madison Logic Seminar, 25 January 2020 1 pm (7 pm UK).

The talk will be held online via Zoom ID: 998 6013 7362.

Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to overcome that criticism—as well as I am able to—by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.

The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.

Set-theoretic and arithmetic potentialism: the state of current developments, CACML 2020

This will be a plenary talk for the Chinese Annual Conference on Mathematical Logic (CACML 2020), held online 13-15 November 2020. My talk will be held 14 November 17:00 Beijing time (9 am GMT).

Potentialist perspectives

Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspective—a statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.

Continuous models of arithmetic, MOPA, November 2020

This will be a talk for the Models of Peano Arithmetic (MOPA) seminar on 11 November 2020, 12 pm EST (5pm GMT). Kindly note the rescheduled date and time.

Abstract. Ali Enayat had asked whether there is a model of Peano arithmetic (PA) that can be represented as $\newcommand\Q{\mathbb{Q}}\langle\Q,\oplus,\otimes\rangle$, where $\oplus$ and $\otimes$ are continuous functions on the rationals $\Q$. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. The reals $\mathbb{R}$, the reals in any finite dimension $\mathbb{R}^n$, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open.

This is joint work with Ali Enayat, myself and Bartosz Wcisło.

Article: Topological models of arithmetic

Modal model theory as mathematical potentialism, Oslo online Potentialism Workshop, September 2020

This will be a talk for the Oslo potentialism workshop, Varieties of Potentialism, to be held online via Zoom on 23 September 2020, from noon to 18:40 CEST (11am to 17:40 UK time). My talk is scheduled for 13:10 CEST (12:10 UK time). Further details about access and registration are availavle on the conference web page.

Abstract. I shall introduce and describe the subject of modal model theory, in which one studies a mathematical structure within a class of similar structures under an extension concept, giving rise to mathematically natural notions of possibility and necessity, a form of mathematical potentialism. We study the class of all graphs, or all groups, all fields, all orders, or what have you; a natural case is the class $\text{Mod}(T)$ of all models of a fixed first-order theory $T$. In this talk, I shall describe some of the resulting elementary theory, such as the fact that the $\mathcal{L}$ theory of a structure determines a robust fragment of its modal theory, but not all of it. The class of graphs illustrates the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_\omega$, $\beth_\omega$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. When augmented with the actuality operator @, modal graph theory becomes fully bi-interpretable with truth in the set-theoretic universe. This is joint work with Wojciech Wołoszyn.

Categorical cardinals, CUNY Set Theory Seminar, June 2020

This will be an online talk for the CUNY Set Theory Seminar, Friday 26 June 2020, 2 pm EST = 7 pm UK time. Contact Victoria Gitman for Zoom access. 

Abstract: Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$ in his 1930 quasi-categoricity result asserting that the models of $\text{ZFC}_2$ are precisely those isomorphic to a rank-initial segment $V_\kappa$ of the cumulative set-theoretic universe $V$ cut off at an inaccessible cardinal $\kappa$. I shall discuss the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the $V_\kappa$ universes are categorically characterized by their sentences or theories. For example, if $\kappa$ is the smallest inaccessible cardinal, then up to isomorphism $V_\kappa$ is the unique model of $\text{ZFC}_2$ plus the sentence “there are no inaccessible cardinals.” This cardinal $\kappa$ is therefore an instance of what we call a first-order sententially categorical cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of $\text{ZFC}_2$ by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford).

The theory of infinite games, including infinite chess, Talk Math With Your Friends, June 2020

This will be accessible online talk about infinite chess and other infinite games for the Talk Math With Your Friends seminar, June 18, 2020 4 pm EST (9 pm UK).  Zoom access information.  Please come talk math with me!

Abstract. I will give an introduction to the theory of infinite games, with examples drawn from infinite chess in order to illustrate various concepts, such as the transfinite game value of a position.

See more of my posts on infinite chess.