Nonlinearity and illfoundedness in the hierarchy of consistency strength and the question of naturality, Italy (AILA), September 2022

This will be a talk for the meeting of The Italian Association for Logic and its Applications (AILA) in Caserta, Italy 12-15 September 2022.

Abstract. Set theorists and philosophers of mathematics often point to a mystery in the foundations of mathematics, namely, that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Why should it be? The phenomenon is thought to carry profound significance for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories, the “one road upward.” 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, however, are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to rebut that criticism 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.

Nonlinearity in the hierarchy of large cardinal consistency strength

This is currently a draft version only of my article-in-progress on the topic of linearity in the hierarchy of consistency strength, especially with large cardinals. Comments are very welcome, since I am still writing the article. Please kindly send me comments by email or just post here.

This article will be the basis of the Weeks 7 & 8 discussion in the Graduate Philosophy of Logic seminar I am currently running with Volker Halbach at Oxford in Hilary term 2021.

I present instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength—as natural or as nearly natural as I can make them—and consider philosophical aspects of the question of naturality with regard to this phenomenon.

It is a mystery often mentioned in the foundations of mathematics, a fundamental phenomenon to be explained, 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 will prove the consistency of the other.

Why should it be linear? Why should the large cardinal notions line up like this, when they often arise from completely different mathematical matters? Measurable cardinals arise from set-theoretic issues in measure theory; Ramsey cardinals generalize ideas in graph coloring combinatorics; compact cardinals arise with compactness properties of infinitary logic. Why should these disparate considerations lead to principles that are linearly related by direct implication and consistency strength?

The phenomenon is viewed by many in the philosophy of mathematics as significant in our quest for mathematical truth. In light of Gödel incompleteness, after all, we must eternally seek to strengthen even our best and strongest theories. Is the linear hierarchy of consistency strength directing us along the elusive path, the “one road upward” as John Steel describes it, toward the final, ultimate mathematical truth? That is the tantalizing possibility.

Meanwhile, we do know as a purely formal matter that the hierarchy of consistency strength is not actually well-ordered—it is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features, however, are weird self-referential assertions constructed in the Gödelian manner via the fixed-point lemma—logic-game trickery, often dismissed as unnatural.

Many set theorists claim that amongst the natural assertions, consistency strengths remain linearly ordered and indeed well ordered. H. Friedman refers to “the apparent comparability of naturally occurring logical strengths as one of the great mysteries of [the foundations of mathematics].” Andrés Caicedo says,

It is a remarkable empirical phenomenon that we indeed have comparability for natural theories. We expect this to always be the case, and a significant amount of work in inner model theory is guided by this belief.

Stephen G. Simpson writes:

It is striking that a great many foundational theories are linearly ordered by <. Of course it is possible to construct pairs of artificial theories which are incomparable under <. However, this is not the case for the “natural” or non-artificial theories which are usually regarded as significant in the foundations of mathematics. The problem of explaining this observed regularity is a challenge for future foundational research.

John Steel writes “The large cardinal hypotheses [the ones we know] are themselves wellordered by consistency strength,” and he formulates what he calls the “vague conjecture” asserting that

If T is a natural extension of ZFC, then there is an extension H axiomatized by large cardinal hypotheses such that T ≡ Con H. Moreover, ≤ Con is a prewellorder of the natural extensions of ZFC. In particular, if T and U are natural extensions of ZFC, then either T ≤ Con U or U ≤ Con T.

Peter Koellner writes

Remarkably, it turns out that when one restricts to those theories that “arise in nature” the interpretability ordering is quite simple: There are no descending chains and there are no incomparable elements—the interpretability ordering on theories that “arise in nature” is a wellordering.

Let me refer to this position as the natural linearity position, the assertion that all natural assertions of mathematics are linearly ordered by consistency strength. The strong form of the position, asserted by some of those whom I have cited above, asserts that the natural assertions of mathematics are indeed well-ordered by consistency strength. By all accounts, this view appears to be widely held in large cardinal set theory and the philosophy of set theory.

Despite the popularity of this position, I should like in this article to explore the contrary view and directly to challenge the natural linearity position.

Main Question. Can we find natural instances of nonlinearity and illfoundedness in the hierarchy of consistency strength?

I shall try my best.

You have to download the article to see my candidates for natural instances of nonlinearity in the hierarchy of large cardinal consistency strength, but I can tease you a little by mentioning that there are various cautious enumerations of the ZFC axioms which actually succeed in enumerating all the ZFC axioms, but with a strictly weaker consistency strength than the usual (incautious) enumeration. And similarly there are various cautious versions of the large cardinal hypothesis, which are natural, but also incomparable in consistency strength.

(Please note that it was Uri Andrews, rather than Uri Abraham, who settled question 16 with the result of theorem 17. I have corrected this from an earlier draft.)

A review of the Gödel fixed-point lemma, with generalizations and applications

This bried unpublished note (11 pages) contains an overview of the Gödel fixed-point lemma, along with several generalizations and applications, written for use in the Week 3 lecture of the Graduate Philosophy of Logic seminar that I co-taught with Volker Halbach at Oxford in Hilary term 2021. The theme of the seminar was self-reference, truth, and consistency strengths, and in this lecture we discussed the nature of Gödel’s fixed-point lemma and generalizations, with various applications in logic.

Contents

1. Introduction
2. Gödel’s fixed-point lemma
An application to the Gödel incompleteness theorem
3. Finite self-referential schemes
An application to nonindependent disjunctions of independent sentences
4. Gödel-Carnap fixed point lemma
Deriving the double fixed-point lemma as a consequence
An application to the provability version of Yablo’s paradox
5. Kleene recursion theorem
An application involving computable numbers
An application involving the universal algorithm
An application to Quine programs and Ouroborous chains
6. References

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).

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.

A new proof of the Barwise extension theorem, and the universal finite sequence, Barcelona Set Theory Seminar, 28 October 2020

This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.

Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1 definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism.  This work is joint with Kameryn J. Williams.

The $\Sigma_1$-definable universal finite sequence

• J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” Journal of Symbolic Logic, 2021.
[Bibtex]
@ARTICLE{HamkinsWilliams2021:The-universal-finite-sequence,
author = {Joel David Hamkins and Kameryn J. Williams},
title = {The $\Sigma_1$-definable universal finite sequence},
journal = {Journal of Symbolic Logic},
year = {2021},
volume = {},
number = {},
pages = {},
month = {},
note = {},
abstract = {},
keywords = {},
eprint = {1909.09100},
archivePrefix = {arXiv},
primaryClass = {math.LO},
source = {},
doi = {10.1017/jsl.2020.59},
}

Abstract. 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, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) 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 $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

• The universal algorithm,
• J. D. Hamkins and H. W. Woodin, “The universal finite set,” Mathematics ArXiv, p. 1–16, 2017.
[Bibtex]
@ARTICLE{HamkinsWoodin:The-universal-finite-set,
author = {Joel David Hamkins and W. Hugh Woodin},
title = {The universal finite set},
journal = {Mathematics ArXiv},
year = {2017},
volume = {},
number = {},
pages = {1--16},
month = {},
note = {Manuscript under review},
abstract = {},
keywords = {under-review},
source = {},
doi = {},
eprint = {1711.07952},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/the-universal-finite-set},
}
• The modal logic of arithmetic potentialism,
• J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” Mathematics ArXiv, p. 1–35, 2018.
[Bibtex]
@ARTICLE{Hamkins:The-modal-logic-of-arithmetic-potentialism,
author = {Joel David Hamkins},
title = {The modal logic of arithmetic potentialism and the universal algorithm},
journal = {Mathematics ArXiv},
year = {2018},
volume = {},
number = {},
pages = {1--35},
month = {},
eprint = {1801.04599},
archivePrefix = {arXiv},
primaryClass = {math.LO},
note = {Under review},
url = {http://wp.me/p5M0LV-1Dh},
abstract = {},
keywords = {under-review},
source = {},
doi = {},
}
• A new proof of the Barwise extension theorem
• Kameryn’s blog post about the paper

Computational self-reference and the universal algorithm, Queen Mary University of London, June 2019

This will be a talk for the Theory Seminar for the theory research group in Theoretical Computer Science at Queen Mary University of London. The talk will be held 4 June 2019 1:00 pm, ITL first floor.

Abstract. Curious, often paradoxical instances of self-reference inhabit deep parts of computability theory, from the intriguing Quine programs and Ouroboros programs to more profound features of the Gödel phenomenon. In this talk, I shall give an elementary account of the universal algorithm, showing how the capacity for self-reference in arithmetic gives rise to a Turing machine program $e$, which provably enumerates a finite set of numbers, but which can in principle enumerate any finite set of numbers, when it is run in a suitable model of arithmetic. In this sense, every function becomes computable, computed all by the same universal program, if only it is run in the right world. Furthermore, the universal algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. An analogous result holds in set theory, where Woodin and I have provided a universal locally definable finite set, which can in principle be any finite set, in the right universe, and which can furthermore be successively extended to become any desired finite superset of that set in a suitable top-extension of that universe.

The modal logic of potentialism, ILLC Amsterdam, May 2019

This will be a talk at the Institute of Logic, Language and Computation (ILLC) at the University of Amsterdam for events May 11-12, 2019. See Joel David Hamkins in Amsterdam 2019.

Abstract: Potentialism can be seen as a fundamentally model-theoretic notion, in play for any class of mathematical structures with an extension concept, a notion of substructure by which one model extends to another. Every such model-theoretic context can be seen as a potentialist framework, a Kripke model whose modal validities one can investigate. In this talk, I’ll explain the tools we have for analyzing the potentialist validities of such a system, with examples drawn from the models of arithmetic and set theory, using the universal algorithm and the universal definition.