A model with no magic sets

On Russell typicality in Set Theory

By Tzouvaras, a set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD$\subseteq$HNT$\subseteq$V holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of ZFC is presented in which HOD$\subsetneqq$HNT$\subsetneqq$V, and another model of ZFC, in which HNT does not satisfy the axiom of choice.

On The Computability of Perfect Subsets of Sets with Positive Measure

A set $X \subseteq 2^ω$ with positive measure contains a perfect subset. We study such perfect subsets from the viewpoint of computability and prove that these sets can have weak computational strength. Then we connect the existence of perfect subsets of sets with positive measure with reverse mathematics.

The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We let \mathbb{R}_{\ind} be the structure with underlying set \mathbb{R} and expanded by all sets of the form st(X), where X \subseteq R^{n} is definable in R and n=1,2,.... We show that the subsets of \mathbb{R}^n that are definable in \mathbb{R}_{\ind} are exactly the finite unions of sets of the form st(X) \setminus st(Y), where X,Y \subseteq R^n are definable…

F-finite embeddabilities of sets and ultrafilters

Let $S$ be a semigroup, let $n\in\mathbb{N}$ be a positive natural number, let $A,B\subseteq S$, let $\mathcal{U},\mathcal{V}\inβS$ and let let $\mathcal{F}\subseteq\{f:S^{n}\rightarrow S\}$. We say that $A$ is $\mathcal{F}$-finitely embeddable in $B$ if for every finite set $F\subseteq A$ there is a function $f\in\mathcal{F}$ such that $f\left(A^{n}\right)\subseteq B$, and we say that $\mathcal{U}$ is $\mathcal{F}$-finitely embeddable in $\mathcal{V}$ if for every set $B\in\mathcal{V}$ there is a set $A\in\mathcal{U}$ such that $A$ is $\mathcal{F}$-finitely embeddable in $B$. We show that $\m…

Automata and tame expansions of $(\mathbb{Z},+)$

The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}$ whose set of representations base $d$ is recognized by a finite automaton, a necessary condition is found for $x+y\in A$ to be a stable formula in $\operatorname{Th}(\mathbb{Z},+,A)$. Combined with a theorem of Moosa and Scanlon this gives a combinatorial characterization of the $d$-automatic $A\subseteq \mathbb{Z}$ such that $(\mathbb{Z},+,A)$ is stable. This characterization is in terms of what were called "$F$-sets" by Moosa and Scanlon …

Strong measure zero sets without Cohen reals

If ZFC is consistent, then each of the following are consistent with ZFC + 2^{aleph_0}= aleph_2 : 1.) X subseteq R is of strong measure zero iff |X| <= aleph_1 + there is a generalized Sierpinski set. 2.) The union of aleph_1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size aleph_2.

A ganzstellensatz for semi-algebraic sets in real closed valued fields

Let $(K,ν)$ be a real closed valued field, and let $S\subseteq K^n$ be a definable open semi-algebraic set. We find an algebraic characterization of rational functions which are OVF-integral on $S$. We apply the existing model theoretic framework for proving ganzstellensätze, and need to control semi-sections and their relations to orderings. (joint work with Noa Lavi)

Kőnig 性质从有限边集的遗传性

Heritability of Kőnig's Property from finite edge sets

针对 Aharoni 关于 Kőnig 性质能否从有限边子集遗传的问题，本文构造反例给出否定回答，并进一步考察更弱性质的遗传性。

低维平面集的钉距

Pinned distances of planar sets with low dimension

本文研究低维平面集的钉距集合，给出其豪斯多夫维数的改进下界，并证明弱正则及紧致Ahlfors-David正则集存在小通用集，使得钉距维数达到最小值。

Stability and sparsity in sets of natural numbers

Given a set $A\subseteq\mathbb{N}$, we consider the relationship between stability of the structure $(\mathbb{Z},+,0,A)$ and sparsity of the set $A$. We first show that a strong enough sparsity assumption on $A$ yields stability of $(\mathbb{Z},+,0,A)$. Such sets include examples considered by Palacín and Sklinos and Poizat, many classical linear recurrence sequences (e.g. the Fibonaccci numbers), and any set in which the limit of ratios of consecutive elements diverges. Finally, we consider sparsity conclusions on sets $A\subseteq\mathbb{N}$, which follow from model theoretic assumptions on $…

Pointwise compact and stable sets of measurable functions

In a series of papers, M.Talagrand, the second author and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the theory of Pettis integration, were solved subject to various special axioms. It was left unclear just how far the special axioms were necessary. In particular, several results depended on the fact that it is consistent to suppose that every countable relatively pointwise compact set of Lebesgue measurable functions is `stable' in Talagrand's sense; the point b…