arXiv subjects

Trevor Wilson

Publications and source records attributed to Trevor Wilson.

6 recordsLinked to original sources

arXiv2026

Model theory of class-sized logics

We study compactness and L\"owenheim-Skolem properties of fragments of the class-sized logic $\mathcal{L}_{\infty \infty}$ and of class-sized versions of second-order and sort logics. In these fragments, certain combinations of infinitary quantifiers and boolean connectives are banned. While model-theoretic properties fail for unrestricted class logics, this drastically changes in our more restricted setting. We show that model-theoretic properties of class logics characterise a wide array of large cardinals, and that some of them can even be obtained in ZFC. In particular, we give a characterisation of Weak Vop\v{e}nka's Principle and Ord is Woodin by downwards L\"owenheim-Skolem properties, and a characterisation of Shelah cardinals by a compactness property of class-sized logics. We further strengthen many known results about properties of set-sized logics by studying how they transfer to class-sized extensions.

math.LO↗

arXiv2026

More Derived Models in PFA

This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $\kappa$. In particular, using a concept called Covering Matrices, we show that the $\Theta$ of the derived model at $\kappa$ is strictly less than $\kappa^+$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $\kappa$ is a limit of Woodin cardinals of cofinality $\omega$ and the derived model does not satisfy $LSA$. Assuming a form of mouse capturing, we show that the derived model satisfies $AD_{\mathbb{R}}$ under $PFA$ when $\kappa$ is a regular limit of Woodin cardinals. If $\kappa$ is an indestructibly $(\kappa,\kappa^+)$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $AD_{\mathbb{R}}$.

math.LO↗

arXiv2025

A model of the Axiom of Determinacy in which every set of reals is universally Baire

The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.'' The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$``there is a cardinal $\lambda$ that is a limit of Woodin cardinals and of $\mathord{<}\lambda$-strong cardinals.'' The $\Sigma^2_1$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel.

math.LO↗

arXiv2022

Ideals and Strong Axioms of Determinacy

We show that the following two theories are equiconsistent: (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal such that if j is the generic embedding associated with it then its restriction on ordinals is independent of the generic object is". (S) ZF, ADR and "Theta is a regular cardinal." The main result of this paper is that T implies that the minimal model of S exists. Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem, establishes the equiconsistency of T and S. Our main result partially resolves a well-known conjecture of Woodin, and completely solves one of the main Core Model Induction problems dating back to 90s.

math.LO↗

arXiv2020

The Weak Vopěnka Principle for definable classes of structures

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures (WVP), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular we show that WVP for $Σ_2$-definable classes is equivalent to the existence of a strong cardinal. The main theorem shows, more generally, that WVP for $Σ_n$-definable classes is equivalent to the existence of a $Σ_n$-strong cardinal. Hence, WVP is equivalent to the existence of a $Σ_n$-strong cardinal, all $n <ω$.

math.LO↗

arXiv2016

Determinacy from strong compactness of $ω_1$

In the absence of the Axiom of Choice, the "small" cardinal $ω_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that $ω_1$ is $X$-strongly compact (where $X$ is any set) if there is a fine, countably complete measure on $\mathcal{P}_{ω_1}(X)$. Working in $\mathsf{ZF} + \mathsf{DC}$, we prove that the $\mathcal{P}(ω_1)$-strong compactness and $\mathcal{P}(\mathbb{R})$-strong compactness of $ω_1$ are equiconsistent with $\mathsf{AD}$ and $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$ respectively, where $\mathsf{AD}$ denotes the Axiom of Determinacy and $\mathsf{AD}_\mathbb{R}$ denotes the Axiom of Real Determinacy. The $\mathcal{P}(\mathbb{R})$-supercompactness of $ω_1$ is shown to be slightly stronger than $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$, but its consistency strength is not computed precisely. An equiconsistency result at the level of $\mathsf{AD}_\mathbb{R}$ without $\mathsf{DC}$ is also obtained.

math.LO↗