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James Cummings

Publications and source records attributed to James Cummings.

At least 19 recordsLinked to original sources
arXiv2026

Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem

We continue the study of compactness phenomena between the set-theoretic universe and $\mathrm{HOD}$ initiated by Goldberg--Poveda \cite{GolPov}. We focus on compactness phenomena around the power-set functions of $V$ and $\mathrm{HOD}$. We prove: (1) A singular strong limit cardinal with uncountable cofinality cannot be the first place where $\mathcal{P}(\cdot )$ and $ \mathcal{P}^{\mathrm{HOD}}(\cdot)$ disagree. (2) Assuming the existence of a measurable cardinal, $\aleph_\omega$ can be the first place where $\mathcal{P}(\aleph_\omega)\neq \mathcal{P}^{\mathrm{HOD}}(\aleph_\omega)$, answering a question of Hayut. (3) If $\kappa$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to $\kappa^+$ and the GCH holds in $\mathrm{HOD}$ below $\kappa^+$ then $(\mathrm{HOD}, V)$ has the $\mathrm{cf}(\kappa)^+$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.

math.LO↗
arXiv2025

Squares, scales and lines

We use hypotheses from PCF theory to construct a linear ordering which has cardinality the successor of a singular cardinal of countable cofinality, and is incompact in the following sense: the ordering is not sigma-scattered, but every smaller subordering is sigma-wellordered. Such orderings were first constructed by Todorcevic using Jensen's square principle.

math.LO↗
arXiv2025

The tree property on long intervals of regular cardinals

In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.

math.LO↗
arXiv2025

Isomorphism Classes of Generating Sets

We introduce a new class of ultrafilters which generalizes the well-known class of simple $P$-point ultrafilters. We prove that for any well-founded $\sigma$-directed partial order $\mathbb{D}$ there is a mild forcing extension where there is an ultrafilter $U$ on $\omega$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. On a measurable cardinal we prove a similar result: relative to a supercompact cardinal, it is consistent that $\kappa$ is supercompact, and for a $\kappa^+$-directed well-founded poset $\mathbb{D}$, there is a ${<}\kappa$-directed closed $\kappa^+$-cc forcing extension where there is a \emph{normal} ultrafilter $U$ on $\kappa$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. These are optimal results in the class of $P$-points and realize every potential structure of a $P$-point. We apply our constructions to obtain ultrafilters with controlled Tukey-type, in particular, an ultrafilter with non-convex Tukey and depth spectra is presented, answering questions from \cite{Benhamou_2024}. Our construction also provides new models where $\mathfrak{u}_\kappa<2^\kappa$, answering questions from \cite{Benhamou_Goldberg2025}.

math.LO↗
arXiv2024

Will you donate money to a chatbot? The effect of chatbot anthropomorphic features and persuasion strategies on willingness to donate

This work investigates the causal mechanism behind the effect of chatbot personification and persuasion strategies on users' perceptions and donation likelihood. In a 2 (personified vs. non-personified chatbot) x 2 (emotional vs. logical persuasion strategy) between-subjects experiment (N=76), participants engaged with a chatbot that represented a non-profit charitable organization. The results suggest that interaction with a personified chatbot evokes perceived anthropomorphism; however, it does not elicit greater willingness to donate. In fact, we found that commonly used anthropomorphic features, like name and narrative, led to negative attitudes toward an AI agent in the donation context. Our results showcase a preference for non-personified chatbots paired with logical persuasion appeal, emphasizing the significance of consistency in chatbot interaction, mirroring human-human engagement. We discuss the importance of moving from exploring the common scenario of a chatbot with machine identity vs. a chatbot with human identity in light of the recent regulations of AI systems.

cs.HC↗
arXiv2023

On minimal non-$σ$-scattered linear orders

The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-$σ$-scattered. Specifically, we will show that Jensen's principle $\diamondsuit$ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-$σ$-scattered linear orders of cardinality greater than $\aleph_1$, as given a successor cardinal $κ^+$, we obtain such linear orderings of cardinality $κ^+$ with the additional property that their square is the union of $κ$-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-$σ$-scattered linear orders of cardinality $κ^+$ exist for every cardinal $κ$ in Gödel's constructible universe, and also (using work of Rinot) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal $μ$ of Mitchell order $μ^{++}$.

math.LO↗
arXiv2019

Ultrafilters on singular cardinals of uncountable cofinality

We prove that consistently there is a singular cardinal $κ$ of uncountable cofinality such that $2^κ$ is weakly inaccessible, and every regular cardinal strictly between $κ$ and $2^κ$ is the character of some uniform ultrafilter on $κ$.

math.LO↗
arXiv2016

A Framework for Forcing Constructions at Successors of Singular Cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size is less than $2^κ$ and such that any graph on $κ^+$ embeds into one of the graphs in the collection.

math.LO↗
arXiv2016

Collapsing the cardinals of $HOD$

Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.

math.LO↗
arXiv2014

Limits, Regularity and Removal for Finite Structures

Our work builds on known results for k-uniform hypergraphs including the existence of limits, a Regularity Lemma and a Removal Lemma. Our main tool here is a theory of measures on ultraproduct spaces which establishes a correspondence between ultraproduct spaces and Euclidean spaces. First we show the existence of a limit object for convergent sequences of relational structures and as a special case, we retrieve the known limits for graphs and digraphs. Then we extend this notion to finite models of a fixed universal theory. We also state and prove a Regularity Lemma and a Removal Lemma. We will discuss connections between our work and Razborov's flag algebras as well.

math.LO↗
arXiv2014

Small universal families of graphs on $\aleph_{ω+1}$

We prove that it is consistent that $\aleph_ω$ is strong limit, $2^{\aleph_ω}$ is large and the universality number for graphs on $\aleph_{ω+1}$ is small. The proof uses Prikry forcing with interleaved collapsing.

math.LO↗
arXiv2012

Singular cardinals and strong extenders

We investigate the circumstances under which there exist a singular cardinal $μ$ and a short $(κ, μ)$-extender $E$ witnessing "$κ$ is $μ$-strong", such that $μ$ is singular in $\Ult(V, E)$.

math.LO↗
arXiv2012

Monochromatic triangles in three-coloured graphs

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.

math.CO↗
arXiv2006

Large cardinals with few measures

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of lambda strong compactness or lambda supercompactness measures on P_kappa(lambda) can be exactly lambda+, if lambda>kappa is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

math.LO↗
arXiv1997

Some independence results on reflection

We prove that there is a certain degree of independence between stationary reflection phenomena at different cofinalities; e.g. it is consistent that every stationary subset of S_1^3 reflects at a point of cofinality aleph_2 while every stationary subset of S^3_0 has a non-reflecting stationary subset

math.LO↗
arXiv1995

A model in which every infinite Boolean algebra has many subalgebras

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}. This implies in particular that B has 2^{|B|} subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a ``black box'' at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.

math.LO↗
arXiv1995

Cardinal invariants above the continuum

We prove some consistency results about b(lambda) and d(lambda), which are natural generalisations of the cardinal invariants of the continuum b and d. We also define invariants b_cl(lambda) and d_cl(lambda), and prove that almost always b(lambda) = b_cl(lambda) and d(lambda)=d_cl(lambda)

math.LO↗