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The $λ$-PSP at $λ$-coanalytic sets

Fernando Barrera, Vincenzo Dimonte, Sandra Müller

TL;DR

The paper investigates when all $λ$-coanalytic subsets of ${}^{λ}2$ have the $λ$-PSP in the setting of a strong limit $λ$ with $cf(λ)=ω$, and proves a staged consistency-strength result: from this assumption one can derive the existence of inner models with increasingly large large-cardinal strength, culminating in $λ$-many measurables. The authors develop a fine-grained core-model framework across $K^{DJ}$, $L[U]$, and Koepke's core model $K[U_{can}]$, and translate regularity properties into inner-model consequences via detailed coding of premice and their iterability, leveraging covering properties and Prikry-type forcing arguments. The work systematically raises the lower bound for the consistency strength of all $λ$-coanalytic sets having the $λ$-PSP, clarifying the role of core-model technology in generalized descriptive set theory at singular, countable cofinality. It thus contributes a staged, tool-rich approach to consistency-strength arguments in GDST and highlights precise thresholds where inner-model existence follows from regularity properties at ${}^{λ}2$.

Abstract

Given a strong limit cardinal $λ$ of countable cofinality, we show that if every $λ$-coanalytic subset of the generalised Cantor space ${}^λ2$ has the $λ$-$\mathsf{PSP}$, then there is an inner model with $λ$-many measurable cardinals. The paper, a contribution to the ongoing research on generalised regularity properties in generalised descriptive set theory at singular cardinals of countable cofinality, is aimed at descriptive set theorists, and so it presents the main result in small steps, slowly increasing the consistency strength lower bound from the existence of an inner model with a measurable cardinal up to the already mentioned one. For this, the inner model theory and covering properties of the Dodd-Jensen core model, $L[U]$ and Koepke's canonical model are used. By giving as much detail as needed at any step, we intend to provide the community with the necessary tools to deal with consistency strength arguments at the corresponding levels.

The $λ$-PSP at $λ$-coanalytic sets

TL;DR

The paper investigates when all -coanalytic subsets of have the -PSP in the setting of a strong limit with , and proves a staged consistency-strength result: from this assumption one can derive the existence of inner models with increasingly large large-cardinal strength, culminating in -many measurables. The authors develop a fine-grained core-model framework across , , and Koepke's core model , and translate regularity properties into inner-model consequences via detailed coding of premice and their iterability, leveraging covering properties and Prikry-type forcing arguments. The work systematically raises the lower bound for the consistency strength of all -coanalytic sets having the -PSP, clarifying the role of core-model technology in generalized descriptive set theory at singular, countable cofinality. It thus contributes a staged, tool-rich approach to consistency-strength arguments in GDST and highlights precise thresholds where inner-model existence follows from regularity properties at .

Abstract

Given a strong limit cardinal of countable cofinality, we show that if every -coanalytic subset of the generalised Cantor space has the -, then there is an inner model with -many measurable cardinals. The paper, a contribution to the ongoing research on generalised regularity properties in generalised descriptive set theory at singular cardinals of countable cofinality, is aimed at descriptive set theorists, and so it presents the main result in small steps, slowly increasing the consistency strength lower bound from the existence of an inner model with a measurable cardinal up to the already mentioned one. For this, the inner model theory and covering properties of the Dodd-Jensen core model, and Koepke's canonical model are used. By giving as much detail as needed at any step, we intend to provide the community with the necessary tools to deal with consistency strength arguments at the corresponding levels.

Paper Structure

This paper contains 9 sections, 49 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GDST at countable cardinals of countable cofinality
  4. Coding structures on ${}^{\lambda}2$
  5. A toy example
  6. The $\lambda{\hbox{-}}\mathsf{PSP}$ up to the existence of an inner model with a measurable cardinal
  7. The $\lambda{\hbox{-}}\mathsf{PSP}$ up to the existence of $0^{\dagger}$
  8. The $\lambda{\hbox{-}}\mathsf{PSP}$ up to the existence of an inner model with $\lambda$-many measurable cardinals
  9. Final remarks and open questions

Key Result

Proposition 2.2

Let $\lambda$ be a singular cardinal of countable cofinality and suppose that $A\subseteq V_{\lambda}$. Then:

Theorems & Definitions (83)

  • Definition 2.1: DMR
  • Proposition 2.2
  • Proposition 2.3: DMR
  • Lemma 2.4: Boundedness Lemma DMR
  • Definition 2.5: DMR
  • Proposition 2.6: DMR
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 73 more