Searcharxiv⌕ Search

arXiv subjects

Ralf Schindler

Publications and source records attributed to Ralf Schindler.

At least 19 recordsLinked to original sources
arXiv2026

The transcendence degree of the reals over certain set-theoretical subfields

It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set $X$ of finitely many Cohen reals, by showing that, in the forcing extension, the transcendence degree of the reals over a combination of the reals in the extension given by each proper subset of $X$ is also maximal. This answers a question of Kanovei and Schindler.

math.LO↗
arXiv2025

Varsovian models II

Assume the existence of sufficent large cardinals. Let $M_{\mathrm{sw}n}$ be the minimal iterable proper class $L[E]$ model satisfying "there are $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that the $δ_i$ are Woodin cardinals and the $κ_i$ are strong cardinals". Let $M=M_{\mathrm{sw}2}$. We identify an inner model $\mathscr{V}_2^M$ of $M$, which is a proper class model satisfying "there are 2 Woodin cardinals", and is iterable both in $V$ and in $M$, and closed under its own iteration strategy. The construction also yields significant information about the extent to which $M$ knows its own iteration strategy. We characterize the universe of $\mathscr{V}_2^M$ as the mantle and the least ground of $M$, and as $\mathrm{HOD}^{M[G]}$ for $G\subseteq\mathrm{Coll}(ω,λ)$ being $M$-generic with $λ$ sufficiently large. These results correspond to facts already known for $M_{\mathrm{sw}1}$, and the proofs are an elaboration of those, but there are substantial new issues and new methods used to handle them.

math.LO↗
arXiv2024

Martin's Maximum${}^{\ast, ++}_{\mathfrak{c}}$ in $\mathbb{P}_{\max}$ extensions of strong models of determinacy

We study a strengthening of $\mathrm{MM}^{++}$ which is called $\mathrm{MM}^{\ast, ++}$ and which was introduced by Asperó and Schindler. We force its bounded version $\mathrm{MM}^{\ast, ++}_{\mathfrak{c}}$, which is stronger than both $\mathrm{MM}^{++}(\mathfrak{c})$ as well as $\mathrm{BMM}^{++}$, by $\mathbb{P}_{\max}$ forcing over a determinacy model $L^{F_{\mathrm{uB}}}({\mathbb R}^*,\mbox{Hom}^{\ast})$. The construction of the ground model $L^{F_{\mathrm{uB}}}({\mathbb R}^{\ast},\mbox{Hom}^{\ast})$ builds upon Gappo and Sargsyan, and the derived model construction of Larson, Sargsyan, and Wilson.

math.LO↗
arXiv2023

PFA and the definability of the nonstationary ideal

We produce, relative to a ${\sf ZFC}$ model with a supercompact cardinal, a ${\sf ZFC}$ model of the Proper Forcing Axiom in which the nonstationary ideal on $ω_1$ is $Π_1$-definable in a parameter from $H_{\aleph_2}$.

math.LO↗
arXiv2022

Coding over Core Models

Early in their careers, both Peter Koepke and Philip Welch made major contributions to two important areas of set theory, core model theory and coding, respectively. In this article we aim to survey some of the work that has been done which combines these two themes, extending Jensen's original Coding Theorem from $L$ to core models witnessing large cardinal properties.

math.LO↗
arXiv2022

Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$

We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $ω_1$ can not be defined by a $Σ_1$ formula with parameter $A \subset ω_1$. We show that the same conclusion holds under the assumption of Woodin's $(\ast)$-axiom. We further show that there are universes where $\BPFA$ holds and $\NS$ is $Σ_1(ω_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\NS$ is saturated, $Σ_1(ω_1)$-definable and $\MA$ holds.

math.LO↗
arXiv2021

Tall cardinals in extender models

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $λ$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $κ$ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.

math.LO↗
arXiv2021

${\sf MM}^{++}$ implies $(*)$

We show that Martin's Maximum${}^{++}$ implies Woodin's ${\mathbb P}_{\rm max}$ axiom $(*)$. This answers a question from the 1990's and amalgamates two prominent axioms of set theory which were both known to imply that there are $\aleph_2$ many real numbers.

math.LO↗
arXiv2019

Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

We prove the following result which is due to the third author. Let $n \geq 1$. If $\boldsymbolΠ^1_n$ determinacy and $Π^1_{n+1}$ determinacy both hold true and there is no $\boldsymbolΣ^1_{n+2}$-definable $ω_1$-sequence of pairwise distinct reals, then $M_n^\#$ exists and is $ω_1$-iterable. The proof yields that $\boldsymbolΠ^1_{n+1}$ determinacy implies that $M_n^\#(x)$ exists and is $ω_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the statement that $\boldsymbolΠ^1_{n+1}$ determinacy implies $\Game^{(n)}(<ω^2 - \boldsymbolΠ^1_1)$ determinacy.

math.LO↗
arXiv2018

When is a real generic over $L$?

In this paper we isolate a new criterion for when a given real $x$ is generic over $L$ in terms of $x$'s capability of lifting elementary embeddings of initial segments of $L$.

math.LO↗
arXiv2018

Inner-model reflection principles

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $φ(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $φ(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $Π_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.

math.LO↗
arXiv2018

The consistency strength of the perfect set property for universally Baire sets of reals

We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver indiscernible is virtually Shelah in $L$. We also show that the statement $\text{uB} = {\bfΔ}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $Σ_2$-reflecting virtually Shelah cardinal.

math.LO↗
arXiv2017

$Σ_1(κ)$-definable subsets of $\mathrm{H}(κ^+)$

We study $Σ_1(ω_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $Σ_1$-formula with parameter $ω_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $Σ_1(ω_1)$-definable, the set of all stationary subsets of $ω_1$ is not $Σ_1(ω_1)$-definable and the complement of every $Σ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$ is not $Σ_1(ω_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a $Σ_1(ω_1)$-definable well-ordering of $\mathrm{H}({ω_2})$ and the existence of a $Δ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no $Σ_1(ω_1)$-definable uniformization of the club filter on $ω_1$. Moreover, we prove a perfect set theorem for $Σ_1(ω_1)$-definable subsets of ${}^{ω_1}ω_1$, assuming that there is a measurable cardinal and the non-stationary ideal on $ω_1$ is saturated. The proofs of these results use iterated generic ultrapowers and Woodin's $\mathbb{P}_{\mathrm{max}}$-forcing. Finally, we also prove variants of some of these results for $Σ_1(κ)$-definable subsets of ${}^κκ$, in the case where $κ$ itself has certain large cardinal properties.

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↗
arXiv2015

Harrington's principle over higher order arithmetic

Let $Z_2$, $Z_3$, and $Z_4$ denote $2^{\rm nd}$, $3^{\rm rd}$, and $4^{\rm th}$ order arithmetic, respectively. We let Harrington's Principle, {\sf HP}, denote the statement that there is a real $x$ such that every $x$--admissible ordinal is a cardinal in $L$. The known proofs of Harrington's theorem "$Det(Σ_1^1)$ implies $0^{\sharp}$ exists" are done in two steps: first show that $Det(Σ_1^1)$ implies {\sf HP}, and then show that {\sf HP} implies $0^{\sharp}$ exists. The first step is provable in $Z_2$. In this paper we show that $Z_2 \, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC}$ and that $Z_3\, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC} \, +$ there exists a remarkable cardinal. As a corollary, $Z_3\, + \, {\sf HP}$ does not imply $0^{\sharp}$ exists, whereas $Z_4\, + \, {\sf HP}$ does. We also study strengthenings of Harrington's Principle over $2^{\rm nd}$ and $3^{\rm rd}$ order arithmetic.

math.LO↗