arXiv subjects

Connor Watson

Publications and source records attributed to Connor Watson.

4 recordsLinked to original sources

arXiv2026

The Cone Fixing the Reals of $\mathrm{HOD}^{M_{2n}(x)}$

For a cone of reals $x$, the reals of $\mathrm{HOD}^{L[x]}$ are precisely the reals in $Q_3$, or equivalently, the set of reals in $M_1$, the minimal proper class mouse with a Woodin cardinal. We show that if $z$ is any real coding $M_1|δ^{+M_1}$, then $z$ is the base of such a cone (so $M_1^{\#}$ is such a base). We also establish a kind of optimality for this result, by identifying a real $x$ such that $M_1\subseteq L[x]$ but $\mathbb{R}\cap\mathrm{HOD}^{L[x]}\not\subseteq M_1$, although there is a base for the cone in a generic extension of $L[x]$. We prove an analogous result for the reals of $\mathrm{HOD}^{M_{n}(x)}$ for even $n$, as well as an analogous optimality result which holds for all $n$.

math.LO↗

arXiv2026

Indiscernible Extraction in Theories with Bounded $k$-splitting

Hanson introduced the class of theories with bounded $k$-splitting, a higher-arity notion of stability, and proved an indiscernible extraction result for those theories assuming the existence of a $k$-ineffable cardinal. In this article, we show that the conclusion of Hanson's theorem is equivalent to the weaker assertion that there exists a weakly compact cardinal, as well as prove additional indiscernible extraction results for these theories in ZFC. Finally, we show that Hanson's construction of total bounded ultraimaginary Morley sequences in theories with bounded $k$-splitting follows from ZFC.

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

Existence of a Model of $o(κ)=κ^{++}$ from Failure of GCH at a Measurable Cardinal

It is well-known that the consistency strength of the GCH failing at a measurable cardinal is the existence of a cardinal $κ$ with $o(κ)=κ^{++}$. As the literature does not contain more than a proof sketch of the lower bound of this equiconsistency, we give an expository proof which fills in the details in order to fill this gap in the literature.

math.LO↗