arXiv subjects

Charles Morgan

Publications and source records attributed to Charles Morgan.

8 recordsLinked to original sources

arXiv2026

Some contributions to presheaf model theory, II -- back and forth

We discuss the back and forth technique in the context of presheaf model theory. The essence of the back and forth technique lies in showing the relationship between various hierarchies which calibrate similarity between two models and, more generally, between two pairs consisting of a model and a tuple from it. In this paper we define several such hierarchies for presheaf models (and tuples of sections from them): those based on the degree of extendibility of partial isomorphisms through literal back and forth conditions, on sharing specific, abstract invariants which we define (the $F^\alpha_{M,\bar{a}}$ of \S{}\ref{function_analysis} for example), on agreeing on the (truth) values of instantiations of formulae up to a given amount of quantifier completity, on the existence of winning strategies for player II in certain Ehrenfeucht-Fra\"iss\'e-type games and, finally, on satisfying certain infinitary sentences that arise in the construction of Scott sentences. We ultimately show that all of these hierarchies align.

math.LO↗

arXiv2026

Scott-Karp analysis without sentences

Scott and Karp gave an analysis which provides a level-by-level equivalence between global similarity between two structures and local commonality in terms of sharing particular invariants. Scott and Karp's local invariants were certain infinitary formulae. We give a more abstract version of the local side of Scott-Karp analysis which avoids the use of infinitary languages. We show the resulting hierarchies are still provide desired equivalences in the classical setting. Moreover, the abstract nature of our analysis, as we show, makes it suitable to provide local level-by-level equivalents to Hjorth's much more general version of global similarity in the context of topological group actions on topological groups. We, furthermore, provide, analogously to the classical work of Ehrenfeucht and Fra\"iss\'e, novel game theoretic equivalents for Hjorth's global similarity relations and some natural variants.

math.LO↗

arXiv2024

Some contributions to presheaf model theory

This paper makes contributions to ``pure'' sheaf model theory, the part of model theory in which the models are sheaves over a complete Heyting algebra. We start by outlining the theory in a way we hope is readable for the non-specialist. We then give a careful treatment of the interpretation of terms and formulae. This allows us to prove various preservation results, including strengthenings of the results of \cite{BM14}. We give refinements of Miraglia's work on directed colimits, \cite{M88}, and an analogue of Tarski's theorem on the preservation of $\forall_2$-sentences under unions of chains. We next show various categories whose objects are (pairs of) presheaves and sheaves with various notions of morphism are accessible in the category theoretic sense. Together these ingredients allow us ultimately to prove that these categories are encompassed in the AECats framework for independence relations developed by Kamsma in \cite{K22}.

math.LO↗

arXiv2024

On forcing axioms and weakenings of the Axiom of Choice

We prove forcing axiom equivalents of two families of weakenings of the axiom of choice: a trichotomy principle for cardinals isolated by L\'evy, ${\rm H\hskip0.05pt}_\kappa$, and ${\rm DC}_\kappa$, the principle of dependent choices generalized to cardinals $\kappa$, for regular cardinals $\kappa$. Using these equivalents we obtain new forcing axiom formulations of the axiom of choice. A point of interest is that we use a new template for forcing axioms. For the class of forcings to which we asks that the axioms apply, we do not ask that they apply to all collections of dense sets of a certain cardinality, but rather only for each particular forcing to a specific family of dense sets of the cardinality in question.

math.LO↗

arXiv2019

Ultrafilters on singular cardinals of uncountable cofinality

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

math.LO↗

arXiv2014

Mitchell-style forcing, with small working parts and collections of models as side conditions, and gap-one simplified morasses

We give a modification of Mitchell's technique for adding objects of size $ω_2$ with conditions with finite working parts in which the collections of models used as side conditions are very highly structured, arguably making them more wieldy. We use one such forcing (essentially a `pure side conditions' forcing) to answer affirmatively the question, asked independently by Shelah and Velleman in the late 1980s, as to whether a $(κ^+,1)$-simplified morass can be added by a forcing with working parts of size $<κ$.

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↗

arXiv2014

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 $\kappa$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $\kappa^+$ whose size is less than $2^\kappa$ and such that any graph on $\kappa^+$ embeds into one of the graphs in the collection.

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