arXiv subjects

David Schrittesser

Publications and source records attributed to David Schrittesser.

At least 19 recordsLinked to original sources

arXiv2026

Loeb Equivalence for General Internal Probability Spaces

Loeb measure theory stands as one of the most influential concepts in nonstandard analysis, underpinning nearly all applications in probability, stochastic processes, and mathematical economics. The paper resolves a fundamental open problem in Loeb measure theory originally posed by Keisler and Sun: let $(Ω,\mathcal{F},μ)$ and $(Ω,\mathcal{G},ν)$ be two Loeb equivalent internal probability spaces, and $\mathcal H$ be the internal algebra generated from $\mathcal{F}\cup\mathcal{G}$. Does there exist an internal probability measure $P$ on $\mathcal H$ such that $(Ω,\mathcal{H},P)$ is Loeb equivalent to $(Ω,\mathcal{F},μ)$? While arXiv:2112.13955 recently provided a positive answer for hyperfinite probability spaces, the problem remained open for general internal probability spaces. We establish the existence of such an internal probability measure for all internal probability spaces.

math.LO↗

arXiv2026

Generalized almost disjoint families and injective Banach spaces

A fundamental open problem in the homological theory of Banach spaces is the calculation of the injective dimension of the Banach space $c_0$. We make a contribution to the study of this problem by proving that, if the Continuum Hypothesis ($\mathsf{CH}$) holds, then the injective dimension of $c_0$ is at least 3. In the course of proving this result, we introduce the notion of an \emph{almost disjoint family} on a topological space $X$, generalizing the classical notion of almost disjoint families of subsets of $\mathbb{N}$, which we feel is of interest in its own right. We prove that, if $\mathfrak{b} = 2^{\aleph_0}$, then there exists an almost disjoint family of cardinality $2^{\aleph_1}$ on the Čech-Stone remainder of $\mathbb{N}$.

math.FA↗

arXiv2025

Definable discrete sets with large continuum

Let $\mathcal R$ be a $Σ^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a $Δ^1_2$ maximal $\mathcal R$-discrete set, and thus the existence of such sets is compatible with the negation of the continuum hypothesis. As an application we find a $Π^1_1$ maximal orthogonal family of Borel probability measures in said extension. The basis of this is a new Ramsey theoretic result.

math.LO↗

arXiv2025

The happy coexistence of mad families and Laver measurability

Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $Π^1_1$ infinite mad family. Since $Π^1_1$ and $Σ^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical pointclasses $Γ$, namely $Γ=Π^1_1$ and $Γ=Σ^1_2$, where $Γ$-uniformization and ``all sets in $Γ$ are Laver measurable'' hold, but there is a mad family in $Γ$. This result stands in contrast to that for reasonable pointclasses, the $Γ$-Ramsey property together with uniformization implies that there are no mad families in $Γ$.

math.LO↗

arXiv2025

Cofinitary groups and projective well-orders

We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideal. Introducing a new robust coding technique, we establish the relative consistency of $\mathfrak{a}_g=\mathfrak{d}<\mathfrak{c}=\aleph_2$ alongside the existence of a $Δ^1_3$-wellorder of the reals and a co-analytic witness for $\mathfrak{a}_g$.

math.LO↗

arXiv2025

Good projective witnesses

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and $\mathfrak{c}$, and there is a $Π^1_2$-definable MCG of this cardinality. Here $Π^1_2$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

math.LO↗

arXiv2024

Maximal eventually different families for uniformly weak Ramsey ideals

We study $\mathcal I$-maximal eventually different families of functions from the set of natural numbers into itself where $\mathcal I$ is an arbitrary ideal on the set of natural numbers that includes the ideal of all finite sets $\mathrm{fin}$. We introduce the class of uniformly weak Ramsey ideals and prove that there exists a closed $\mathcal I$-maximal eventually different family if $\mathcal I$ belongs to this class; this is the case for arbitrary $F_σ$ ideals and Fubini products $\mathrm{fin}^α$ with $α<ω_1$.

math.LO↗

arXiv2024

The Ramsey property and higher dimensional mad families

We prove that under a principle of Ramsey regularity there are no infinite maximal almost disjoint families with respect to the transfinitely iterated Fréchet ideals. The results of the present paper were announced by the authors in the Proceedings of the National Academy of Sciences of the U.S.A.

math.LO↗

arXiv2024

Two-Person Adversarial Games are Zero-Sum: An Elaboration of a Folk Theorem

The observation that every two-person adversarial game is an affine transformation of a zero-sum game is traceable to Luce & Raiffa (1957) and made explicit in Aumann (1987). Recent work of (ADP) Adler et al. (2009), and of Raimondo (2023) in increasing generality, proves what has so far remained a conjecture. We present two proofs of an even more general formulation: the first draws on multilinear utility theory developed by Fishburn & Roberts (1978); the second is a consequence of the ADP proof itself for a special case of a two-player game with a set of three actions.

econ.TH↗

arXiv2023

de Finetti's theorem and the existence of regular conditional distributions and strong laws on exchangeable algebras

We show the following generalizations of the de Finetti--Hewitt--Savage theorem: Given an exchangeable sequence of random elements, the sequence is conditionally i.i.d. if and only if each random element admits a regular conditional distribution given the exchangeable $σ$-algebra (equivalently, the shift invariant or the tail algebra). We use this result, which holds without any regularity or technical conditions, to demonstrate that any exchangeable sequence of random elements whose common distribution is Radon is conditional iid.

math.PR↗

arXiv2022

Statistical minimax theorems via nonstandard analysis

For statistical decision problems with finite parameter space, it is well-known that the upper value (minimax value) agrees with the lower value (maximin value). Only under a generalized notion of prior does such an equivalence carry over to the case infinite parameter spaces, provided nature can play a prior distribution and the statistician can play a randomized strategy. Various such extensions of this classical result have been established, but they are subject to technical conditions such as compactness of the parameter space or continuity of the risk functions. Using nonstandard analysis, we prove a minimax theorem for arbitrary statistical decision problems. Informally, we show that for every statistical decision problem, the standard upper value equals the lower value when the $\sup$ is taken over the collection of all internal priors, which may assign infinitesimal probability to (internal) events. Applying our nonstandard minimax theorem, we derive several standard minimax theorems: a minimax theorem on compact parameter space with continuous risk functions, a finitely additive minimax theorem with bounded risk functions and a minimax theorem on totally bounded metric parameter spaces with Lipschitz risk functions.

math.ST↗

arXiv2022

Definability of maximal cofinitary groups

We present a proof of a result, previously announced by the second author, that there is a closed (even $Π^0_1$) set generating an $F_σ$ (even $Σ^0_2$) maximal cofinitary group (short, mcg) which is isomorphic to a free group. In this isomorphism class, this is the lowest possible definitional complexity of an mcg.

math.GR↗

arXiv2022

Constructing maximal cofinitary groups

Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in $\textsf{ZF}$, that is, without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second order arithmetic with only a few natural number quantifiers.

math.LO↗

arXiv2022

A co-analytic Cohen indestructible maximal cofinitary group

Assuming that every set is constructible, we find a $Π^1_1$ maximal cofinitary group of permutations of $\mathbb N$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans' result that there exists a $Π^1_1$ maximal cofinitary group in $L$.

math.LO↗

arXiv2022

Definable maximal discrete sets in forcing extensions

Let $\mathcal R$ be a $Σ^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a $Δ^1_2$ maximal $\mathcal{R}$-discrete set. We use this to answer in the negative the main question posed in \cite{Fischer2010} by showing that in the Sacks and Miller extensions there is a $Π^1_1$ maximal orthogonal family ("mof") of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over $L$ then there are no $Σ^1_2$ mofs.

math.LO↗

arXiv2022

Projective measure without projective Baire

We prove that it is consistent (relative to a Mahlo cardinal) that all projective sets of reals are Lebesgue measurable, but there is a $Δ^1_3$ set without the Baire property. The complexity of the set which provides a counterexample to the Baire property is optimal.

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

Lightface $Σ^1_2$-indescribable cardinals

$Σ^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{ω_1}}^V \prec_{Σ_2}{H_{ω_1}}^{V^P}$. "$ω_1$ inaccessible to reals" means that for any real $r$, ${ω_1}^{L[r]}<ω_1$. To measure the exact consistency strength of "$Σ^1_3$-absoluteness for ccc forcing and $ω_1$ is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $Σ^1_2$-indescribable cardinal; $κ$ has this property exactly if it is inaccessible and $H_κ\prec_{Σ_2} H_{κ^+}$.

math.LO↗