arXiv subjects

John Steel

Publications and source records attributed to John Steel.

7 recordsLinked to original sources

arXiv2023

$\Sigma_1$ gaps as derived models and correctness of mice

Assume ZF + AD + V=L(R). Let $[\alpha,\beta]$ be a $\Sigma_1$ gap with $J_\alpha(R)$ admissible. We analyze $J_\beta(R)$ as a natural form of "derived model" of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_\beta(R)=\mathcal{P}(R)\cap D$, and if $J_\beta(R)\models$ "$\Theta$ exists", then $J_\beta(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R))^M$, for $\omega$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper.

math.LO↗

arXiv2022

Full normalization for mouse pairs

We develop the theory of meta-iteration trees, that is, iteration trees whose base "model" is itself an ordinary iteration tree. We prove a comparison theorem for meta-iteration strategies parallel to the one for ordinary iteration strategies, and use it to show that the iteration strategy component of a mouse pair condenses to itself under weak tree embeddings. These constitute a class of embeddings between iteration trees that is significantly larger than the class of embeddings mentioned in the definition of mouse pair. We then use this very strong hull condensation property of mouse pairs to show that every iterate of a mouse pair is an iterate via a single $\lambda$-tight, normal iteration tree, and that the associated tail strategies are independent of how the iterate was reached.

math.LO↗

arXiv2022

Suslin cardinals and cutpoints in mouse limits

We obtain a partial result on the following conjecture. Conjecture. Let (P, {\Sigma}) be a projectum stable mouse pair, and let \kappa be a cardinal of V such that \kappa < o(M_\infty(P, {\Sigma})); then the following are equivalent: (1) \kappa is a Suslin cardinal, (2) \kappa is a cutpoint of M_\infty(P, {\Sigma}).

math.LO↗

arXiv2022

Condensation for Mouse Pairs

In this paper, we prove a fine condensation theorem. This is quite similar to condensation theorems for pure extender mice in the literature, except that condensation for iteration strategies has been added to the mix.

math.LO↗

arXiv2021

The mouse set conjecture for sets of reals

Recall that the Mouse Set Conjecture says that under AD++V=L(P(R)), a real is ordinal definable if and only if it belongs to an iterable mouse. The Mouse Set Conjecture for sets of reals says that under the same theory, a set of reals is ordinal definable from a real if and only if it belongs to a mouse over the reals. We prove that the Mouse Set Conjecture implies the Mouse Set Conjecture for sets of reals.

math.LO↗

arXiv2015

Square principles in Pmax extensions

By forcing with $\mathbb{P}_{\rm max}$ over strong models of determinacy, we obtain models where different square principles at $ω_2$ and $ω_3$ fail. In particular, we obtain a model of $2^{\aleph_0}=2^{\aleph_1}=\aleph_2 + \lnot\square(ω_2) + \lnot\square(ω_3)$.

math.LO↗

arXiv2012

Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research.

math.LO↗