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James E. Hanson

Publications and source records attributed to James E. Hanson.

17 recordsLinked to original sources
arXiv2026

Generically stable Keisler measures

Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $μ$ in $T$, we show that the following conditions are equivalent: $(i)$ $μ$ is a frequency interpretation measure; $(ii)$ $μ$ is definable and its canonical "random extension" $r_μ$ is generically stable in the randomization theory $T^R$; $(iii)$ $μ$ is "self-averaging". This result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\Rightarrow(ii)\Rightarrow (iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\Rightarrow (ii)\Rightarrow(i)$. We also prove that generically stable measures are closed under Morley products, answering another well-known question that was open even in the case of types. These results are obtained through the use of AI models.

math.LO↗
arXiv2026

Every $\mathrm{AW}^*$-Algebra is Normal

Using set-theoretic methods, we prove that every $\mathrm{AW}^*$-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of $\mathrm{AW}^*$-factors, by work of Saitô and Wright. We also show that if there is a model $V$ of $\mathrm{ZFC}$ with an $\mathrm{AW}^*$-algebra that fails to be monotone complete, then in some forcing extension of $V$ there is an $\mathrm{AW}^*$-factor that fails to be monotone complete, implying that any $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-factors are monotone complete yields a $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-algebras are monotone complete. Both results build on transfer principles for Boolean-valued factor representations originally developed by Ozawa. This work was assisted by the Danus LLM orchestration system.

math.OA↗
arXiv2026

The Axiom of Double Complement and its opposites

Powell introduced the Axiom of Double Complement ($\mathsf{DCom}$) to give his double-negation interpretation of $\mathsf{ZF}$ into $\mathsf{IZF_{Rep}}$. However, the consistency, strength, and compatibility of $\mathsf{DCom}$ remain open problems. This article aims to survey the compatibility and consistency strength of $\mathsf{DCom}$, its consequence and opposites, which will be named $\mathsf{NDCom}$ and $\mathsf{ADCom}$. We will also develop Lubarsky's Kripke models over $\mathsf{CZF}$ to derive these results. We will show that $\mathsf{DCom}$ proves the Powerset axiom over $\mathsf{CZF}$ and is independent of $\mathsf{IZF}$. We will also show that $\mathsf{ADCom}$ does not add consistency strength over $\mathsf{CZF}$, by modifying the construction of Lubarsky's model for $\mathsf{CZF+\lnot Pow}$. We will also show that $\mathsf{DCom}$, $\mathsf{ADCom}$, and $\mathsf{NDCom}$ are persistent under realizability under modest conditions.

math.LO↗
arXiv2026

Set theory, logic, and homeomorphism groups of manifolds

We investigate the relationship between axiomatic set theory and the first-order theory of homeomorphism groups of manifolds in the language of group theory, concentrating on first-order rigidity and type versus conjugacy. We prove that under the axiom of constructibility (i.e.~{V=L}), homeomorphism groups of arbitrary connected manifolds are first-order rigid, and that the conjugacy class of a homeomorphism of a manifold is determined by its type. In contradistinction, under the regularity hypothesis that every projective set of reals has the Baire property, we show that in all dimensions greater than one there exist pairs of noncompact, connected manifolds whose homeomorphism groups are elementarily equivalent but which are not homeomorphic. We also show, under the same Baire-property hypothesis, that every manifold of positive dimension admits pairs of homeomorphisms with the same type which are not conjugate to each other. Projective determinacy implies the Baire-property hypothesis, so the corresponding consequences under PD follow immediately. Finally, we show that infinitary formulas do determine conjugacy classes of homeomorphisms and homeomorphism types of manifolds; specifically, the conjugacy class of a homeomorphism of an arbitrary manifold is determined by a single $L_{ω_1ω}$ formula. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{ω_1ω}$ sentence.

math.GT↗
arXiv2026

A natural haystack of differentially closed fields

In this partially expository paper, we present a novel construction of differentially closed fields of characteristic $0$: Let $\mathcal{K}_{\mathrm{dense}}$ be the differential ring of all meromorphic functions whose domain is a (not necessarily connected) dense open subset of $\mathbb{C}$ modulo agreement on dense open sets (i.e., $f$ and $g$ are considered equal if there is a dense open $U \subseteq \mathbb{C}$ such that $f|_U = g|_U$). We show that every ring ideal of $\mathcal{K}_{\mathrm{dense}}$ is a differential ideal and that for every maximal ideal $\mathfrak{m}$, the quotient $\mathcal{K}_{\mathrm{dense}}/\mathfrak{m}$ is a differentially closed field. We also show that $\mathcal{K}_{\mathrm{dense}}/\mathfrak{m}$ is saturated and has cardinality of the continuum, implying that any two such quotients are isomorphic as differential fields. We then discuss how to motivate this construction in terms of set-theoretic forcing, Boolean-valued models, and $\neg\neg$-sheaves on $\mathbb{C}$, taking the opportunity to present an impressionistic expository account of these ideas. Finally, we discuss some immediate generalizations of this construction involving the real and $p$-adic numbers and ask some questions about them.

math.LO↗
arXiv2026

Pointwise Mean Value Theorems in Constructive Mathematics

We answer some questions regarding the mean value theorem and related results in constructive mathematics. The answers to these questions reveal interesting properties of the Mean Value Theorem, Law of Bounded Change, and Constancy Principle. We see that, in contrast to the Intermediate Value Theorem whose approximate analogue was shown to hold constructively by Frank, the natural approximate versions of these theorems fail to hold in neutral constructive mathematics. Our proof of this makes use of the existence of a topos in which the real numbers are in bijection with the naturals, which was shown by Bauer and the second author. Using Booij's notion of locators, we show that the aforementioned approximate analogues do hold in the presence of a small amount of countable choice, and also under suitable locator lifting hypotheses. We also show that an even weaker approximate analogue of the Mean Value Theorem holds in neutral constructive mathematics.

math.LO↗
arXiv2026

The Countable Reals

We construct a topos in which the Dedekind reals are countable. The topos arises from a new kind of realizability, which we call parameterized realizability, based on partial combinatory algebras whose application depends on a parameter. Realizers operate uniformly with respect to a given parameter set. Our construction uses a sequence of reals in $[0,1]$, discovered by Joseph Miller, that is non-diagonalizable in the sense that any real which is oracle-computable uniformly from representations of the sequence must already appear in it. When used as the parameter set, this yields a topos in which the non-diagonalizable sequence becomes an epimorphism onto the Dedekind reals, rendering them internally countable. The resulting topos is intuitionistic: it refutes both the law of excluded middle and countable choice. Nevertheless, much of analysis survives internally. The Cauchy reals are uncountable. The Hilbert cube is countable, so Brouwer's fixed-point theorem follows from Lawvere's. The intermediate value theorem and the analytic form of the lesser limited principle of omniscience hold, while the limited principle of omniscience fails. Although no real-valued map has a jump, it remains open whether all such maps are continuous. Finally, the closed interval $[0,1]$, being countable, can be covered by a sequence of open intervals of total length less than any $ε> 0$, with no finite subcover. Yet, we show that any cover using intervals with rational endpoints must admit a finite subcover.

math.LO↗
arXiv2026

Model theoretic events

We develop a notion of sampling, called \emph{generic sampling}, for the context of global Keisler measures where the standard product is replaced by the Morley product. Choosing a point randomly in this space with respect to our distribution yields a \emph{random generic type} in infinitely many variables. We investigate several natural model-theoretic events and provide conditions under which they occur for almost all random generic types.

math.LO↗
arXiv2026

A formula for any real number, maybe

We discuss how to write down three specific natural numbers $A$, $B$, $C$ such that for any real number $r$ you've probably ever thought of, it is consistent with $\mathsf{ZFC}$ set theory that $$\def\Rb{\mathbb{R}}\def\Nb{\mathbb{N}}r = \log\left(\sup_{x_0,x_1 \in \Rb} \inf_{x_2 \in \Rb} \sup_{x_3 \in \Rb}\inf_{x_4 \in \Rb}\sup_{m \in \Nb}\inf_{n_0,\dots,n_{A} \in \Nb} x^2_0 \begin{bmatrix} \phantom{+}(n_0 - 2)^2 + (n_1-m)^2 \\ + n_2 + (n_B - n_C)^2 \\ + n_3 \sum_{k=0}^4 ( x_k - \frac{n_{k+5}}{1+n_4} +n_4)^2 \\ + \sum_{i,j = 0}^B (n_{9+2^i3^j} - n_i^{n_j})^2 \end{bmatrix} \right).$$ We also discuss why it's possible, assuming the existence of certain large cardinals, for there to be a real number $s$ which cannot be the value of this formula for our particular $A$, $B$, $C$. This involves set-theoretic mice.

math.LO↗
arXiv2025

Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure $μ_{W}$ such that generic sampling with respect to $μ_{W}$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k = 2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph.

math.CO↗
arXiv2025

Labelled growth rates of $ω$-categorical structures and applications in choiceless set theory

We study the labelled growth rate of an $ω$-categorical structure $\mathfrak{A}$, i.e., the number of orbits of $Aut(\mathfrak{A})$ on $n$-tuples of distinct elements, and show that the model-theoretic property of monadic stability yields a gap in the spectrum of allowable labelled growth rates. As a further application, we obtain gap in the spectrum of allowable labelled growth rates in hereditary graph classes, with no a priori assumption of $ω$-categoricity. We also establish a way to translate results about labelled growth rates of $ω$-categorical structures into combinatorial statements about sets with weak finiteness properties in the absence of the axiom of choice, and derive several results from this translation.

math.LO↗
arXiv2025

Any function I can actually write down is measurable, right?

In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert's tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial $p(x,y,z,n,k_1,\dots,k_{70})$ of degree $7$ with integer coefficients such that it is independent of $\mathsf{ZFC}$ (and much stronger theories) whether the function $$f(x) = \inf_{y \in \mathbb{R}}\sup_{z \in \mathbb{R}}\inf_{n \in \mathbb{N}}\sup_{\bar{k} \in \mathbb{N}^{70}}p(x,y,z,n,\bar{k})$$ is Lebesgue measurable. We also give similarly defined $g(x,y)$ with the property that the statement "$x \mapsto g(x,r)$ is measurable for every $r \in \mathbb{R}$" has large cardinal consistency strength (and in particular implies the consistency of $\mathsf{ZFC}$) and $h(m,x,y,z)$ such that $h(1,x,y,z),\dots,h(16,x,y,z)$ can consistently be the indicator functions of a Banach$\unicode{x2013}$Tarski paradoxical decomposition of the sphere. Finally, we discuss some situations in which measurability of analogously defined functions can be concluded by inspection, which touches on model-theoretic o-minimality and the fact that sufficiently strong large cardinal hypotheses (such as Vopěnka's principle and much weaker assumptions) imply that all 'reasonably definable' functions (including the above $f(x)$, $g(x,y)$, and $h(m,x,y,z)$) are universally measurable.

math.LO↗
arXiv2025

Bi-invariant types, reliably invariant types, and the comb tree property

We introduce and examine some special classes of invariant types$\unicode{x2014}$bi-invariant, strongly bi-invariant, extendibly invariant, and reliably invariant types$\unicode{x2014}$and show that they are related to certain model-theoretic tree properties. We show that the comb tree property (recently introduced by Mutchnik) is equivalent to the failure of Kim's lemma for bi-invariant types and is implied by the failure of Kim's lemma for reliably invariant types over invariance bases. We show that every type over an invariance base extends to a reliably invariant type$\unicode{x2014}$generalizing an unpublished result of Kruckman and Ramsey$\unicode{x2014}$and use this to show that, under a reasonable definition of Kim-dividing, Kim-forking coincides with Kim-dividing over invariance bases in theories without the comb tree property. Assuming a measurable cardinal, we characterize the comb tree property in terms of a form of dual local character. We also show that the antichain tree property (introduced by Ahn and Kim) seems to have a somewhat similar relationship to strong bi-invariance. In particular, we show that NATP theories satisfy Kim's lemma for strongly bi-invariant types and (assuming a measurable cardinal) satisfy a different form of dual local character. Furthermore, we examine a mutual generalization of the local character properties satisfied by NTP$_2$ and NSOP$_1$ theories and show that it is satisfied by all NATP theories. Finally, we give some related minor results$\unicode{x2014}$a strengthened local character characterization of NSOP$_1$ and a characterization of coheirs in terms of invariant extensions in expansions$\unicode{x2014}$as well as a pathological example of Kim-dividing.

math.LO↗
arXiv2025

A combinatorial characterization of Kim's lemma for pairs of bi-invariant types

We give a combinatorial consistency-inconsistency configuration that is equivalent to the failure of the following form of Kim's lemma for a given $k$: $(\star)$ For any set of parameters $A$, formula $φ(x,b)$, and $A$-bi-invariant types $p$ and $q$ extending $\mathrm{tp}(b/A)$, if $φ(x,b)$ $k$-divides along $p$, then it divides along $q$. We then give an equivalent technical variant of $(\star)$ that is non-trivial over arbitrary invariance bases. We also show that the failure of weaker versions of $(\star)$ entails the existence of stronger combinatorial configurations, the strongest of which can be phrased in terms of families of parameters indexed by arbitrary cographs (i.e., $P_4$-free graphs). Finally, we show that if there is an array $(b_{i,j} : i,j < ω)$ of parameters such that $\{φ(x,b_{i,j}) : (i,j) \in C\}$ is consistent whenever $C \subseteq ω^2$ is a chain (in the product partial order) and $k$-inconsistent whenever $C$ is an antichain, then there is a model $M$, parameter $b$, and $M$-coheirs $p,q \supset \mathrm{tp}(b/M)$ such that $q^{\otimes ω}$ is an $M$-heir-coheir and $φ(x,b)$ $k$-divides along $p$ but does not divide along $q$. In doing so, we also show that this configuration entails the failure of generic stationary local character under the assumption of $\mathsf{GCH}$.

math.LO↗
arXiv2025

Indiscernible extraction at small large cardinals from a higher-arity stability notion

We introduce a higher-arity stability notion defined in terms of $k$-splitting, a higher-arity generalization of splitting. We show that theories with bounded $k$-splitting have improved indiscernible extraction at $k$-ineffable cardinals, and we give a non-trivial example of a theory with bounded $k$-splitting but unbounded $(k-1)$-splitting for each odd $k > 1$. We also show that bounded $k$-splitting implies $\mathrm{NFOP}_k$, a higher arity stability notion introduced by Terry and Wolf. We then use our indiscernible extraction result together with a construction of Kaplan and Shelah to give a strong counterexample to the converse: an $\mathrm{NIP}$ theory with unbounded $k$-splitting for every $k$. Finally, as a thematically related but technically independent result, we show that treelessness implies $\mathrm{NFOP}_2$, sharpening a result of Kaplan, Ramsey, and Simon.

math.LO↗
arXiv2023

Generic stability, randomizations, and NIP formulas

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations, and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler-Morley measure, which plays the role of a Morley sequence for a Keisler measure. We prove that if $μ$ is fim over $M$, then for any Keisler-Morley measure $λ$ in $μ$ over $M$ and any formula $φ(x,b)$, $\lim_{i \to \infty} λ(φ(x_i,b)) = μ(φ(x,b))$. We also show that any measure satisfying this conclusion must be fam. (2) We study the map, defined by Ben Yaacov, taking a definable measure $μ$ to a type $r_μ$ in the randomization. We prove that this map commutes with Morley products, and that if $μ$ is fim then $r_μ$ is generically stable. (3) We characterize when generically stable types are closed under Morley products by means of a variation of ict-patterns. Moreover, we show that NTP$_2$ theories satisfy this property. (4) We prove that if a local measure admits a suitably tame global extension, then it has finite packing numbers with respect to any definable family. We also characterize NIP formulas via the existence of tame extensions for local measures.

math.LO↗