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A model with fragments of projective determinacy and failures of $\mathsf{DC}$

Sandra Müller, Bartosz Wcisło

TL;DR

The paper establishes that boldface $m{ ext{Pi}}^1_n$-determinacy does not imply $oldsymbol{ ext{Pi}}^1_{n+2}$-$ ext{DC}_{oldsymbol{ ext{R}}}$ by constructing a symmetric extension from an inner-model-theoretic base with Woodin cardinals, where a definable tree of reals has no branch and a particular instance of DC fails. The method adapts the Gitman–Friedman–Kanovei forcing framework, using a carefully designed poset $oldsymbol{ ext{J}}$ built via a diamond-guided sequence of Sacks-type iterations and a seal operation to produce a tree of reals encoding inductive generics. The results are extended to contexts with Woodin cardinals, showing definable failures of DC at higher projective levels while preserving determinacy at an adjacent level in small forcing extensions (via $M_n$ and $M_n^{ iny ext{#}}$ preservation). Overall, the work clarifies the nuanced relationship between projective determinacy and choice principles, demonstrating separations that persist under targeted large-cardinal hypotheses and forcing constructions.

Abstract

We describe a construction of a model of second order arithmetic in which (boldface) $\bm{Π^1_n}$-determinacy holds, but (lightface) $Π^1_{n+2}$-$\mathsf{DC}$ fails, thus showing that no projective level of determinacy implies full $\mathsf{DC}_{\mathbb{R}}$. The construction builds upon the work of Gitman, Friedman, and Kanovei.

A model with fragments of projective determinacy and failures of $\mathsf{DC}$

TL;DR

The paper establishes that boldface -determinacy does not imply - by constructing a symmetric extension from an inner-model-theoretic base with Woodin cardinals, where a definable tree of reals has no branch and a particular instance of DC fails. The method adapts the Gitman–Friedman–Kanovei forcing framework, using a carefully designed poset built via a diamond-guided sequence of Sacks-type iterations and a seal operation to produce a tree of reals encoding inductive generics. The results are extended to contexts with Woodin cardinals, showing definable failures of DC at higher projective levels while preserving determinacy at an adjacent level in small forcing extensions (via and preservation). Overall, the work clarifies the nuanced relationship between projective determinacy and choice principles, demonstrating separations that persist under targeted large-cardinal hypotheses and forcing constructions.

Abstract

We describe a construction of a model of second order arithmetic in which (boldface) -determinacy holds, but (lightface) - fails, thus showing that no projective level of determinacy implies full . The construction builds upon the work of Gitman, Friedman, and Kanovei.

Paper Structure

This paper contains 9 sections, 13 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inner Model Theory
  4. Symmetric models
  5. Adapting Gitman--Friedman--Kanovei poset
  6. A model with failures of $\mathsf{DC}$ and $\Pi^1_1$-determinacy
  7. A measurable cardinal implies determinacy in the symmetric model
  8. Definable failures of $\mathsf{DC}$ in models with Woodin cardinals
  9. Projective determinacy in the small forcing extensions of $M_n$

Key Result

Lemma 1

Let $M^* \subset M[G]$ be a symmetric extension of $M$. Suppose that $X,a \in M^*$ and let $Y$ be a subset of $X$ definable in $M[G]$ with the parameter $a$. Then $Y \in M^*$.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5: Kanovei--Lyubetsky Theorem for $\mathbb{J}$
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 8 more