[Paper Review] Scales in hybrid mice over $\mathbb{R}$

Farmer Schlutzenberg, Nam Trang|arXiv (Cornell University)|Oct 26, 2012

Advanced Topology and Set Theory8 references3 citations

TL;DR

This paper establishes a scales analysis in hybrid mice over ℝ, specifically in the structure $\mathrm{Lp}^{{{}^{\mathrm{G}}}Ω}(\mathbb{R},\Omega\mathord{\upharpoonright}\mathrm{HC})$, by generalizing Steel’s scale constructions in $L(\mathbb{R})$ and $\mathrm{Lp}(\mathbb{R})$ to $\Theta$-g-organized $\Omega$-premice. The key contribution is a systematic scale existence theorem for these hybrid mice under optimal determinacy hypotheses, enabling core model induction to reach stronger inner model-theoretic conclusions.

ABSTRACT

We develop a general theory of strategic mice, prove their condensation properties, and analyze the scales pattern in the stack of $Θ$-g-organized $\mathcal{F}$-mice over $\mathbb{R}$, Lp$^{G\mathcal{F}}(\mathbb{R})$, for a class of nice operators $\mathcal{F}$.

Motivation & Objective

• To extend the theory of scales in inner models beyond $L(\mathbb{R})$ and $\mathrm{Lp}(\mathbb{R})$ to more complex hybrid mice over $\mathbb{R}$.
• To resolve foundational issues in defining $\Sigma$-premice over $\mathbb{R}$ due to the lack of a well-ordering on $\mathbb{R}$.
• To develop a framework for scale construction in $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ using $\Theta$-g-organized $\Omega$-premice and self-scaled parameters $\Upsilon$.
• To provide a foundation for core model induction to derive stronger consistency strength results, such as models of $\mathsf{AD}^{+} + \Theta > \Theta_0$.
• To establish conditions under which scale existence theorems can be applied in the context of core model induction, particularly via super-small mouse capturing and Wadge compatibility.

Proposed method

• Introduce $\Theta$-g-organized $\Omega$-premice as a variant of Sargsyan’s reorganized hod premice, enabling a hierarchy of mice over $\mathbb{R}$ despite the absence of a well-ordering.
• Define $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ as the stack of sound, countably iterable $\Theta$-g-organized $\Omega$-premice projecting to $\mathbb{R}$, where $\Upsilon = \Omega\mathord{\upharpoonright}\mathrm{HC}$ is self-scaled.
• Adopt a three-case framework for scale construction analogous to Steel’s work: (1) when $\mathcal{J}(\mathcal{M}) \models \mathsf{AD}$, (2) when $\mathcal{M}$ ends a weak gap, and (3) when $\mathcal{M}$ is a limit of such models.
• Use optimal determinacy hypotheses and super-small mouse capturing to ensure scale existence in each case, particularly leveraging Wadge compatibility of $\Upsilon^\mathrm{cd}$ with boldface $\mathsf{AD}$-pointclasses.
• Apply the scale existence theorems to core model induction by verifying that the assumptions hold at initial segments of $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$, especially under the condition that $\Omega$ is very nice.
• Establish that $\mathcal{J}(\mathcal{N}) \models \mathsf{AD}$ for relevant $\mathcal{N}$ by showing that $\mathfrak{P}(\mathbb{R}) \cap \mathcal{J}(\mathcal{N}) \subseteq \mathfrak{P}(\mathbb{R}) \cap \mathcal{J}(\mathrm{HC}, \Upsilon)$, which follows from Wadge compatibility and the structure of projective sets in $\mathcal{J}(\mathcal{N})$.

Experimental results

Research questions

• RQ1 How can scale theory be extended from $L(\mathbb{R})$ and $\mathrm{Lp}(\mathbb{R})$ to hybrid mice over $\mathbb{R}$, particularly when $\mathbb{R}$ lacks a well-ordering?
• RQ2 What structural properties of $\Omega$-premice are necessary to define a coherent hierarchy of mice over $\mathbb{R}$, and how can $\Theta$-g-organization help overcome the lack of a well-ordering?
• RQ3 Under what conditions does $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ admit scales, and how can this be established from optimal determinacy assumptions?
• RQ4 How does the scale existence in $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ support core model induction in constructing models of $\mathsf{AD}^{+} + \Theta > \Theta_0$?
• RQ5 What role does Wadge compatibility of $\Upsilon^\mathrm{cd}$ with boldface $\mathsf{AD}$-pointclasses play in ensuring that $\mathcal{J}(\mathcal{N}) \models \mathsf{AD}$ for relevant initial segments $\mathcal{N}$?

Key findings

• The paper establishes a systematic scale existence theorem for $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ under optimal determinacy hypotheses, generalizing Steel’s results in $L(\mathbb{R})$ and $\mathrm{Lp}(\mathbb{R})$.
• The construction of scales in $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ proceeds in three cases: when $\mathcal{J}(\mathcal{M}) \models \mathsf{AD}$, when $\mathcal{M}$ ends a weak gap, and when $\mathcal{M}$ is a limit of such models, with theorems 5.17, 5.22, and 5.26 providing the respective scale constructions.
• It is shown that $\mathcal{J}(\mathcal{N}) \models \mathsf{AD}$ for any $\mathcal{N} \triangleleft \mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ ending a weak gap, provided $\Omega$ is very nice and $\Upsilon^\mathrm{cd}$ is Wadge compatible with all boldface $\mathsf{AD}$-pointclasses.
• The paper proves that $\mathfrak{P}(\mathbb{R}) \cap \mathcal{J}(\mathcal{N}) \subseteq \mathfrak{P}(\mathbb{R}) \cap \mathcal{J}(\mathrm{HC}, \Upsilon)$, which implies $\mathcal{J}(\mathcal{N}) \models \mathsf{AD}$, under the assumption that $\Upsilon$ is self-scaled and $\Omega$ is very nice.
• The scale existence theorems are applied to core model induction, showing that under optimal determinacy and super-small mouse capturing, the scales analysis holds throughout $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$.
• The framework ensures that $\mathrm{Lp}^{{}^{\mathrm{G}}}Ω(\mathbb{R},\Upsilon)$ agrees with $\mathrm{Lp}^{\Omega}(\mathbb{R},\Upsilon)$ on $\mathfrak{P}(\mathbb{R})$ and has identical extender sequences when $\Omega$ relativizes well.