Tag Archives: forcing

The contingent HOD dichotomy, Notre Dame Logic Seminar, September 2026

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This will be a talk for the Logic Seminar at the University of Notre Dame on 15 September 2026, 2pm in the Hayes-Healy building room 125.

The contingent HOD dichotomy

Joel David Hamkins

Abstract: We shall discuss the contingently contingent nature of the class HOD of hereditarily ordinal-definable sets. In some models of set theory the axiom V=HOD is contingent by set forcing and in others it is not. After discussing some philosophical and historical puzzles concerning the nature of ordinal definability, I shall introduce and investigate what we call the contingent HOD dichotomy. Namely, it is provable in ZFC that either (1) the set-theoretic universe V is close to HOD in several respects: HOD is a ground model of V; the axiom V=HOD is forceable by set forcing; every object is ordinal-definable with a single additional parameter; and furthermore all these things are forcing invariant and hold throughout the generic multiverse; or (2) the universe is far from HOD; in particular V ≠ HOD;  more generally, HOD is not a ground; the axiom V=HOD is not forceable; the universe is not ordinal-definable from a parameter; and these things hold invariantly throughout the generic multiverse. We shall explore how the dichotomy engages with large cardinals, with set-theoretic geology, with the iterated HOD^alpha models, with the maximality principles, and with Woodin’s HOD dichotomy. 

This is new joint work in progress with Bokai Yao (Peking University).

The Contingent HOD Dichotomy, CUNY Logic Workshop, October 2026

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This will be a talk for the CUNY Logic Workshop, 23 October 2026, 2pm, at the CUNY Graduate Center in midtown Manhattan.

The Contingent HOD Dichotomy

Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame

Abstract: We shall discuss the contingently contingent nature of the class HOD of hereditarily ordinal-definable sets. In some models of set theory the axiom V=HOD is contingent by set forcing and in others it is not. After discussing some philosophical and historical puzzles concerning the nature of ordinal definability, I shall introduce and investigate what we call the contingent HOD dichotomy. Namely, it is provable in ZFC that either (1) the set-theoretic universe V is close to HOD in several respects: HOD is a ground model of V; the axiom V=HOD is forceable by set forcing; every object is ordinal-definable with a single additional parameter; and furthermore all these things are forcing invariant and hold throughout the generic multiverse; or (2) the universe is far from HOD; in particular V ≠ HOD;  more generally, HOD is not a ground; the axiom V=HOD is not forceable; the universe is not ordinal-definable from a parameter; and these things hold invariantly throughout the generic multiverse. We shall explore how the dichotomy engages with large cardinals, with set-theoretic geology, with the maximality principles, and with Woodin’s HOD dichotomy.

This is new joint work in progress with Bokai Yao (Peking University).

On the contingent contingency of V = HOD and independence over the maximality principles, Fudan University, Shanghai, July 2026

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This will be a talk for the Fudan University logic group on 16 July 4:00pm in Shanghai.

Abstract: The axiom V = HOD is contingently contingent—more precisely, it is class-forcing contingent that V = HOD is contingent with respect to set forcing. It follows that V = HOD and □(V ≠ HOD) are logically independent of the maximality principles MP and MP(ℝ) over ZFC. I will give a full review of the relevant notions. This will be a chalkboard talk concerning very new joint work in progress with Bokai Yao (Peking University).

The paper is in progress and will be available I expect in a few weeks.

The Modal Logic of Forcing and Set-theoretic Potentialism, Peking University lectures, June/July 2026

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This will be a series of graduate lectures at Peking University, two lectures per week beginning mid-June and proceeding into July.

Topics. We shall aim to cover the central results in the modal logic of forcing, including an exploration of the main tools, and then explore how those ideas apply more generally in set-theoretic potentialism and other potentialist contexts, such as arithmetic potentialism and modal model theory. For prerequisites, students should be already familiar with some graduate-level set theory, including the basics of forcing, as well as standard tools from mathematical logic and model theory.

The lectures will focus on various research papers, as follows:

  • Joel David Hamkins. “A simple maximality principle.” Journal of Symbolic Logic 68.2 (2003),pp. 527-550. doi:10.2178/jsl/1052669062. arXiv:math/0009240.
  • Joel David Hamkins and Benedikt Löwe. “The modal logic of forcing.” Trans. AMS 360.4 (2008), pp. 1793-1817. doi:10.1090/S0002-9947-07-04297-3. arXiv:math/0509616.
  • Joel David Hamkins and Øystein Linnebo. “The Modal Logic of Set-theoretic Potentialism and the Potentialist Maximality Principles.” Review of Symbolic Logic 15.1 (2022), pp. 1-35. doi:10.1017/S1755020318000242. arXiv:1708.01644.
  • Joel David Hamkins and Wojciech Aleksander Wołoszyn. “Modal Model Theory.” Notre Dame Journal of Formal Logic 65.1 (2024), pp. 1-37. doi:10.1215/00294527-2024-0001.
  • Joel David Hamkins and Øystein Linnebo. “Second-order Potentialism.” research manuscript in preparatoin.
  • Joel David Hamkins. “The Modal Logic of Arithmetic Potentialism and the Universal Algorithm.” Philosophia Mathematica 34.1 (2026), pp. 137-182. doi:0.1093/philmat/nkag001.
  • Joel David Hamkins. “Every countable model of arithmetic or set theory has a pointwise definable end extension.” Kriterion Journal of Philosophy (2024). doi:10.1515/krt-2023-0029. arXiv:2209.12578.
  • Additional readings may be added, if time permits.

Forcing is simply the iterative conception undertaken with multivalued logic, ForcingFest, Oslo, June 2024

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I shall be speaking at the ForcingFest meeting at the University of Oslo, 21 June 2024.

Abstract. I will explain how the forcing construction can be seen as a direct implementation of the iterative conception, giving rise to the cumulative hierarchy, but undertaken in the context of multivalued logic. The shape of the logic that is available in effect enables a certain constructive interference of the truth values in such a way that can affect the truth judgements. The core utility of forcing arises from the fact that we can often control these consequences by making a careful choice of the logic to be used, thereby controlling the truth values even of natural set-theoretic statements such as the continuum hypothesis.

Slides-ForcingFest-Oslo-Hamkins-June-2024Download

The computable model theory of forcing, Rutgers Logic Seminar, December 2023

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This will be a talk for the Rutgers University Logic Seminar, December 4, 2023.

Abstract. I shall discuss the computable model theory of forcing. To what extent can we view forcing as a computational process on the models of set theory? Given an oracle for the atomic or elementary diagram of a model (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of 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 that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.

The paper is available on the arxiv at https://arxiv.org/abs/2007.00418.

Set-theoretic forcing as a computational process, Midwest Computability Seminar, Chicago, May 2023

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This is a talk for the MidWest Computability Seminar conference held May 2, 2023 at the University of Chicago. The talk will be available via Zoom at https://notredame.zoom.us/j/99754332165?pwd=RytjK1RFZU5KWnZxZ3VFK0g4YTMyQT09.

Abstract: I shall explore several 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 (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of 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 that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.

The paper is available on the arxiv at https://arxiv.org/abs/2007.00418.

The talk took place in “The Barn” in the upper space between the Reyerson Laboratory and Eckhart Hall, where the University of Chicago Department of Mathematics is located:

Slides-Forcing-as-a-computational-process-MCS-Chicago-2023Download

A survey of set-theoretic geology, Notre Dame Logic Seminar, January 2023

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This will be a talk 31 January 2-3 for the Notre Dame Logic Seminar.

Abstract. I shall give a general introduction and account of the main elements of set-theoretic geology, the motivating questions, the central definitions, and the main results, including newer advances. We’ll discuss ground models, the ground axiom, the mantle, the ground-model definability theorem, Usuba’s results on downward directedness and more. 

Pseudo-countable models

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[bibtex key=”Hamkins:Pseudo-countable-models”]

Download pdf at arXiv:2210.04838

Abstract. Every mathematical structure has an elementary extension to a pseudo-countable structure, one that is seen as countable inside a suitable class model of set theory, even though it may actually be uncountable. This observation, proved easily with the Boolean ultrapower theorem, enables a sweeping generalization of results concerning countable models to a rich realm of uncountable models. The Barwise extension theorem, for example, holds amongst the pseudo-countable models—every pseudo-countable model of ZF admits an end extension to a model of ZFC+V=L. Indeed, the class of pseudo-countable models is a rich multiverse of set-theoretic worlds, containing elementary extensions of any given model of set theory and closed under forcing extensions and interpreted models, while simultaneously fulfilling the Barwise extension theorem, the Keisler-Morley theorem, the resurrection theorem, and the universal finite sequence theorem, among others.

The sentence asserting its own non-forceability by nontrivial forcing

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At the meeting here in Konstanz, Giorgo Venturi and I considered the sentence $\sigma$, which asserts its own non-forceability by nontrivial forcing. That is, $\sigma$ asserts that there is no nontrivial forcing notion forcing $\sigma$. $$\sigma\quad\iff\quad \neg\exists\mathbb{B}\ \Vdash_{\mathbb{B}}\sigma.$$ The sentence $\sigma$ would be a fixed-point of the predicate for not being nontrivially forceable.

In any model of set theory $V$ in which $\sigma$ is true, then in light of what it asserts, it would not be forceable by nontrivial forcing, and so it would be false in all nontrivial forcing extensions of that model $V[G]$. And in any model $W$ where it is false, then because of what it asserts, it would be nontrivially forceable, and so it would be true in some forcing extension of that model $W[G]$.

But this is a contradiction! It cannot ever be true, since if it were true in $V$, it would have to be false in all extensions $V[G]$, and therefore true in some subsequent extension $V[G][H]$. But that model is a forcing extension of $V$, contradicting the claim that it is false in all such extensions.

So it must always be false, but this can’t happen, since then in any given model, in light of what it asserts, it would have to be true. So it cannot ever be true or false.

Conclusion: there is no such sentence σ that asserts its own nontrivial forceability. This is no fixed-point for not being nontrivially forceable.

But doesn’t this contradict the fixed-point lemma? After all, the fixed-point lemma shows that we can produce fixed points for any expressible assertion.

The resolution of the conundrum is that although for any given assertion $\varphi$, we can express “$\varphi$ is forceable”, we cannot express “x is the Gödel code of a forceable sentence”, for reasons similar to those for Tarski’s theorem on the nondefinability of truth.

Therefore, we are not actually in a situation to apply the fixed-point lemma. And ultimately the argument shows that there can be no sentence $\sigma$ that asserts “$\sigma$ is not forceable by nontrivial forcing”.

Ultimately, I find the logic of this sentence $\sigma$, asserting its own non-nontrivial forceability, to be a set-theoretic forcing analogue of the Yablo paradox. The sentence holds in a model of set theory whenever it fails in all subsequent models obtained by forcing, and that relation is exactly what arises in the Yablo paradox.

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

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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-Kobe-2021 SlidesDownload

Forcing as a computational process

[bibtex key=”HamkinsMillerWilliams:Forcing-as-a-computational-process”]

Forcing as a computational process

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[bibtex key=”HamkinsMillerWilliams:Forcing-as-a-computational-process”]

Abstract. We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory $\langle M,\in^M\rangle$, we explain senses in which one may compute $M$-generic filters $G\subseteq\mathbb{P}\in M$ and the corresponding forcing extensions $M[G]$. Specifically, from the atomic diagram one may compute $G$, from the $\Delta_0$-diagram one may compute $M[G]$ and its $\Delta_0$-diagram, and from the elementary diagram one may compute the elementary diagram of $M[G]$. We also examine the information necessary to make the process functorial, and conclude that in the general case, no such computational process will be functorial. For any such process, it will always be possible to have different isomorphic presentations of a model of set theory $M$ that lead to different non-isomorphic forcing extensions $M[G]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

Forcing as a computational process, Cambridge, Februrary 2019

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This will be a talk for Set Theory in the United Kingdom (STUK 1), to be held in the other place, February 16, 2019.

Abstract. We investigate the 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, we explain senses in which one may 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.

The rearrangement number: how many rearrangements of a series suffice to validate absolute convergence? Warwick Mathematics Colloquium, October 2018

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This will be a talk for the Mathematics Colloquium at the University of Warwick, to be held October 19, 2018, 4:00 pm in Lecture Room B3.02 at the Mathematics Institute. I am given to understand that the talk will be followed by a wine and cheese reception.Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

Set-theoretic blockchains

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[bibtex key=”HabicHamkinsKlausnerVernerWilliams2018:Set-theoretic-blockchains”]

Abstract. 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 posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.