arXiv subjects

Derek Levinson

Publications and source records attributed to Derek Levinson.

4 recordsLinked to original sources

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

Derived Models in PFA

We discuss a conjecture of Wilson that under the proper forcing axiom, $\Theta_0$ of the derived model at $\kappa$ is below $\kappa^+$. We prove the conjecture holds for the old derived model. Assuming mouse capturing in the new derived model, the conjecture holds there as well. We also show $\Theta < \kappa^+$ in the case of the old derived model, and under additional hypotheses for the new derived model.

math.LO↗

arXiv2023

Unreachability of $\bf{\Gamma_{2n+1,m}}$

We find bounds for the maximal length of a sequence of distinct $\bf{\Gamma_{2n+1,m}}$-sets under $AD$ and show there is no sequence of distinct $\bf{\Gamma_{2n+1}}$-sets of length $\bf{\delta^1_{2n+3}}$. As a special case, there is no sequence of distinct $\bf{\Gamma_{1,m}}$-sets of length $\aleph_{m+2}$. These are the optimal results for the pointclasses $\bf{\Gamma_{2n+1}}$ and $\bf{\Gamma_{1,m}}$.

math.LO↗

arXiv2022

Unreachability of Inductive-Like Pointclasses in $L(\mathbb{R})$

Hjorth proved from $ZF + AD + DC$ that there is no sequence of distinct $\Sigma^1_2$ sets of length $\delta^1_2$. Sargsyan extended Hjorth's technique to show there is no sequence of distinct $\Sigma^1_{2n}$ sets of length $\delta^1_{2n}$. Sargsyan conjectured an analogous property is true for any regular Suslin pointclass in $L(R)$ -- i.e. if $\kappa$ is a regular Suslin cardinal in $L(R)$, then there is no sequence of distinct $\kappa$-Suslin sets of length $\kappa^+$ in $L(R)$. We prove this in the case that the pointclass $S(\kappa)$ is inductive-like.

math.LO↗