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Separating Many Localisation Cardinals on the Generalised Baire Space

Tristan van der Vlugt

Abstract

Given a cofinal cardinal function $h\in{}^κκ$ for $κ$ inaccessible, we consider the dominating $h$-localisation number, that is, the least cardinality of a dominating set of $h$-slaloms such that every $κ$-real is localised by a slalom in the dominating set. It was proved in arXiv:1611.08140 that the dominating localisation numbers can be consistently different for two functions $h$ (the identity function and the power function). We will construct a $κ$-sized family of functions $h$ and their corresponding localisation numbers, and use a ${\leq}κ$-supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above $κ$ is consistent. This answers an open question from arXiv:1611.08140 .

Separating Many Localisation Cardinals on the Generalised Baire Space

Abstract

Given a cofinal cardinal function for inaccessible, we consider the dominating -localisation number, that is, the least cardinality of a dominating set of -slaloms such that every -real is localised by a slalom in the dominating set. It was proved in arXiv:1611.08140 that the dominating localisation numbers can be consistently different for two functions (the identity function and the power function). We will construct a -sized family of functions and their corresponding localisation numbers, and use a -supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above is consistent. This answers an open question from arXiv:1611.08140 .
Paper Structure (2 sections, 15 theorems, 4 equations)

This paper contains 2 sections, 15 theorems, 4 equations.

Table of Contents

  1. The Forcing
  2. Products

Key Result

Lemma 4

${\mathbb S^{h}_\kappa}$ is ${<}\kappa$-closed. That is, for any $\lambda<\kappa$, if $\left\langle T_\xi\mid \xi<\lambda\right\rangle$ is a descending chain of conditions, then there exists a condition $T$ such that $T\leq T_\xi$ for all $\xi<\lambda$.

Theorems & Definitions (17)

  • Definition 2
  • Definition 3
  • Lemma 4
  • Corollary 5
  • Lemma 6
  • Theorem 7
  • Corollary 8
  • Lemma 9
  • Theorem 10
  • Lemma 11
  • ...and 7 more