[Paper Review] The weakness of the pigeonhole principle under hyperarithmetical reductions

Benoît Monin, Ludovic Patey|arXiv (Cornell University)|May 20, 2019

Computability, Logic, AI Algorithms35 references4 citations

TL;DR

This paper establishes that the infinite pigeonhole principle for 2-colorings ($\operatorname{RT}^1_2$) admits strong cone avoidance under hyperarithmetical reductions, meaning that for any non-hyperarithmetical set $B$, every set $A$ has an infinite homogeneous subset $H$ (in $A$ or its complement) such that $B$ remains non-hyperarithmetical relative to $H$. The key contribution is a novel forcing construction generalizing jump control, which also yields strong cone avoidance for $\Delta^0_n$ and arithmetical reductions, resolving a question by Wang on low $n$ subsets.

ABSTRACT

The infinite pigeonhole principle for 2-partitions ($\\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.

Motivation & Objective

• To investigate the computability-theoretic strength of the infinite pigeonhole principle ($\mathrm{RT}^1_2$) under various reducibilities.
• To determine whether $\mathrm{RT}^1_2$ admits strong cone avoidance for $\Delta^0_n$, arithmetical, and hyperarithmetical reductions.
• To resolve a question posed by Wang regarding the existence of low $n$ infinite homogeneous subsets within $\Delta^0_n$ sets.
• To develop a new forcing notion that generalizes the first- and second-jump control techniques of Cholak, Jockusch, and Slaman.

Proposed method

• Introduces a new forcing notion $\mathbb{P}_{\omega_1^{\mathrm{ck}}}$ that generalizes jump control to handle hyperarithmetical reductions.
• Uses a hierarchy of largeness classes $\mathcal{L}_C$ and a notion of $\mathcal{B}_{n,\beta}$-genericity to control the hyperarithmetic complexity of the generic object.
• Employs a recursive construction of conditions to ensure that the generic filter $\mathcal{F}$ does not compute $\omega_1^{\mathrm{ck}}$, thus preserving $\omega_1^H = \omega_1^{\mathrm{ck}}$ for the generic $H$.
• Applies a notion of $\mathcal{B}_{n,\beta}$-genericity to ensure that no $G$-computable function can code $\omega_1^{\mathrm{ck}}$, thereby preventing $\omega_1^G > \omega_1^{\mathrm{ck}}$.
• Uses the fact that $\{G : n \in \mathcal{O}_\alpha^G\}$ is $\Delta^1_1$ uniformly in $\alpha$ and $n$ to define forcing conditions that block hyperarithmetic coding.
• Combines forcing with properties of $\Delta^0_n$-genericity and $\Sigma^1_1$-genericity to prove that $\omega_1^{G_\mathcal{F}} = \omega_1^{\mathrm{ck}}$ for sufficiently generic filters.

Experimental results

Research questions

• RQ1 Does $\mathrm{RT}^1_2$ admit strong cone avoidance for $\Delta^0_n$ reductions?
• RQ2 Does $\mathrm{RT}^1_2$ admit strong cone avoidance for arithmetical reductions?
• RQ3 Does every $\Delta^{0}_{n+1}$-computable set $A$ have an infinite homogeneous subset $H$ of low $n+2$ degree?
• RQ4 Can strong cone avoidance be extended to hyperarithmetical reductions, and does it preserve $\omega_1^H = \omega_1^{\mathrm{ck}}$?
• RQ5 Is it possible to construct a forcing notion that generalizes jump control to handle hyperarithmetical complexity?

Key findings

• The paper proves that $\mathrm{RT}^1_2$ admits strong cone avoidance for $\Delta^0_n$ reductions, meaning that for any non-$\emptyset^{(n)}$-computable set $B$, there exists an infinite homogeneous subset $H$ of $A$ or $\overline{A}$ such that $B$ is not $H^{(n)}$-computable.
• It establishes strong cone avoidance for arithmetical reductions: for any non-arithmetical set $B$, there exists an infinite homogeneous subset $H$ of $A$ or $\overline{A}$ such that $B$ is not arithmetical in $H$.
• It shows that every $\Delta^{0}_{n+1}$-computable set $A$ has an infinite homogeneous subset $H$ of low $n+2$ degree, answering a question of Wang.
• The paper constructs a forcing notion $\mathbb{P}_{\omega_1^{\mathrm{ck}}}$ such that for any sufficiently generic filter $\mathcal{F}$, the generic object $G_{\mathcal{F}}$ satisfies $\omega_1^{G_{\mathcal{F}}} = \omega_1^{\mathrm{ck}}$, ensuring that no $G_{\mathcal{F}}$-computable function can code $\omega_1^{\mathrm{ck}}$.
• It proves that for any non-hyperarithmetical set $B$, there exists an infinite homogeneous subset $H$ of $A$ or $\overline{A}$ such that $B$ is not hyperarithmetical in $H$, with $\omega_1^H = \omega_1^{\mathrm{ck}}$, thus establishing strong cone avoidance for hyperarithmetical reductions.
• The forcing construction ensures that the generic object does not compute $\omega_1^{\mathrm{ck}}$, and hence preserves the $\omega_1$-order type of the constructible hierarchy.