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String Dimension: VC Dimension for Infinite Shattering

Calliope Ryan-Smith

TL;DR

A precise characterisation of such notions of forcing as a generalisation of distributivity is demonstrated, as well as the finality of forcing iteration.

Abstract

In computer science, combinatorics, and model theory, the VC dimension is a central notion underlying far-reaching topics such as error rate for decision rules, combinatorial measurements of classes of finite structures, and neo-stability theory. In all cases, it measures the capacity for a collection of sets $\mathcal{F}\subseteq\mathscr{P}(X)$ to shatter subsets of $X$. The VC dimension of this class then takes values in $\mathbb{N}\cup\{\infty\}$. We extend this notion to an infinitary framework and use this to generate ideals on $2^κ$ of families of bounded shattering. We explore the cardinals characteristics of ideals generated by this generalised VC dimension, dubbed string dimension, and present various consistency results. We also introduce the finality of forcing iteration. A $κ$-final iteration is one for which any sequences of ground model elements of length less than $κ$ in the final model must have been introduced at an intermediate stage. This technique is often used for, say, controlling sets of real numbers when manipulating values of cardinal characteristics, and is often exhibited as a consequence of a chain condition. We demonstrate a precise characterisation of such notions of forcing as a generalisation of distributivity.

String Dimension: VC Dimension for Infinite Shattering

TL;DR

A precise characterisation of such notions of forcing as a generalisation of distributivity is demonstrated, as well as the finality of forcing iteration.

Abstract

In computer science, combinatorics, and model theory, the VC dimension is a central notion underlying far-reaching topics such as error rate for decision rules, combinatorial measurements of classes of finite structures, and neo-stability theory. In all cases, it measures the capacity for a collection of sets to shatter subsets of . The VC dimension of this class then takes values in . We extend this notion to an infinitary framework and use this to generate ideals on of families of bounded shattering. We explore the cardinals characteristics of ideals generated by this generalised VC dimension, dubbed string dimension, and present various consistency results. We also introduce the finality of forcing iteration. A -final iteration is one for which any sequences of ground model elements of length less than in the final model must have been introduced at an intermediate stage. This technique is often used for, say, controlling sets of real numbers when manipulating values of cardinal characteristics, and is often exhibited as a consequence of a chain condition. We demonstrate a precise characterisation of such notions of forcing as a generalisation of distributivity.
Paper Structure (19 sections, 25 theorems, 25 equations, 6 figures)

This paper contains 19 sections, 25 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Model theory
  5. VC dimension
  6. Cardinal characteristics
  7. Forcing
  8. String dimension
  9. Topology
  10. Cardinal characteristics of $\mathcal{I}_{\mathfrak{s}}(\delta,\kappa)$
  11. $\mathfrak{sd}$ for strong limit cardinals
  12. The general construction
  13. Forcing
  14. Finality
  15. Cohen forcing
  16. ...and 4 more sections

Key Result

Theorem 1.2

If $\kappa$ is a strong limit and $\delta$ is the least cardinal such that $2^\delta\geq\mathop{\mathrm{cf}}\nolimits(\kappa)$ then $\mathfrak{sd}(\delta,\kappa)=2^\kappa$. In particular, $\mathfrak{sd}(\aleph_0,\aleph_0)=2^{\aleph_0}$.

Figures (6)

  • Figure 1: The combinatorial cardinal characteristics of $\mathcal{I}$.
  • Figure 2: Cichoń's diagram
  • Figure 3: Relationships between values of $\mathfrak{sd}(\delta,\kappa)$ proved by $\mathsf{ZFC}$.
  • Figure 4: Constellation of $\mathfrak{sd}(\delta,\tau)$ for $\tau=\aleph_0$ or $\aleph_1$ before and after forcing with $\mathop{\mathrm{Add}}\nolimits(\omega_1,\kappa)$.
  • Figure 5: Constellation of $\mathfrak{sd}(\delta,\tau)$ for $\tau=\aleph_0$ or $\aleph_1$ after forcing with $\mathop{\mathrm{Add}}\nolimits(\omega_1,\kappa)\times\mathop{\mathrm{Add}}\nolimits(\omega,\lambda)$.
  • ...and 1 more figures

Theorems & Definitions (66)

  • Definition 1.1: VC class
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1: VC dimension
  • Theorem : Hrushovski--Peterzil--Pillay
  • Theorem : Baldwin--Saxl
  • Definition 2.2: Tree
  • Definition 2.4: Product forcing
  • Definition 2.6: Forcing iteration
  • Definition 3.1: Shattering and string dimension
  • ...and 56 more