Related papers: The higher sharp IV: the higher levels
We establish the descriptive set theoretic representation of the mouse $M_n^{\#}$, which is called $0^{(n+1)\#}$. This part deals with the case $n=1$.
We establish the descriptive set theoretic representation of the mouse $M_n^{\#}$, which is called $0^{(n+1)\#}$. This part partially deals with the case $n=2$ by proving the many-one equivalence of $M_2^{\#}$ and the theory of…
We establish the descriptive set theoretic representation of the mouse $M_n^{\#}$, which is called $0^{(n+1)\#}$. This part partially finishes the case $n=2$ by establishing the higher level analog of the EM blueprint definition of…
In this paper we explore a connection between descriptive set theory and inner model theory. From descriptive set theory, we will take a countable, definable set of reals, A. We will then show that A is equal to the reals of M, where M is a…
Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_\alpha(\mathbb{R})$ where $[\alpha,\alpha]$ is a projective-like gap (of $L(\mathbb{R})$) and $\alpha$ is either a successor ordinal or has…
We identify a particular mouse, $M^{\text{ld}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{\sharp}$ for all $n$, and we show that $\mathbb{R}\cap M^{\text{ld}} = Q_{\omega+1}$, the set of reals that are…
We give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is related to the Martin-Solovay construction for scales on…
Several notions of multiplicativity are introduced for forms of degree $d\geq 3$ over a field of characteristic 0 or greater than d. Examples of multiplicative and strongly multiplicative forms of higher degree are given. Conditions…
We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the…
We present a uniform description of sets of $m$ linear forms in $n$ variables over the field of rational numbers whose computation requires $m(n - 1)$ additions.
We define and study a higher-dimensional version of model theoretic internality, and relate it to higher-dimensional definable groupoids in the base theory.
This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented…
Using ideas from synthetic topology, a new approach to descriptive set theory is suggested. Synthetic descriptive set theory promises elegant explanations for various phenomena in both classic and effective descriptive set theory.…
We study the decomposition as an $\textrm{SO}(3)$-module of the multiplicity space corresponding to the branching from $\textrm{SO}(n+3)$ to $\textrm{SO}(n)$. Here, $\textrm{SO}(n)$ (resp.\ $\textrm{SO}(3)$) is considered embedded in…
Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid…
The paper tries to extend results of the classical Descriptive Set Theory to as many countably based T_0-spaces (cb_0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we…
We classify the maximal $m$-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of the Johnson graph $J(n, m)$ for $m = 2, 3$. Furthermore, we determine the necessary and sufficient condition for $n$ and $m$ such that the…
We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is ageneralization to arbitrary…
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify…
For many applications, we need to use techniques to represent convex shapes and objects. In this work, we use level set method to represent shapes and find a necessary and sufficient condition on the level set function to guarantee the…