[Paper Review] Scales and the fine structure of K(R). Part I: Acceptability above the reals

Daniel W. Cunningham|ArXiv.org|May 16, 2006

Advanced Topology and Set Theory6 references3 citations

TL;DR

This paper establishes the acceptability above the reals for iterable real premice in the inner model $K(\mathbb{R})$, a foundational result enabling the construction of minimal-complexity scales in descriptive set theory. The key contribution is proving that every iterable real premouse $\mathcal{M}$ is acceptable above the reals, which underpins the existence of scales in $K(\mathbb{R})$ and supports subsequent results on boldface pointclasses.

ABSTRACT

This article is Part I in a series of three papers devoted to determining the minimal complexity of scales in the inner model $K(\mathbb{R})$. Here, in Part I, we shall complete our development of a fine structure theory for $K(\mathbb{R})$ which is essential for our work in Parts II and III. In particular, we prove the following fundamental theorem which supports our analysis of scales in $K(\mathbb{R})$: If $\mathcal{M}$ is an iterable real premouse, then $\mathcal{M}$ is acceptable above the reals. This theorem will be used in Parts II and III to solve the problem of finding scales of minimal complexity in $K(\mathbb{R})$.

Motivation & Objective

• To develop a fine structure theory for $K(\mathbb{R})$ that supports the analysis of scales in the inner model.
• To establish the acceptability above the reals for iterable real premice, a critical property for scale construction.
• To resolve foundational questions about the existence of minimal-complexity scales in $K(\mathbb{R})$ under determinacy assumptions.
• To provide the theoretical groundwork for Parts II and III, which analyze boldface pointclasses and scale properties.

Proposed method

• Generalizing Dodd-Jensen's acceptability concept to include the set of reals, defining 'acceptability above the reals' in Definition 3.15.
• Using transfinite recursion on the ordinal height of premice to prove acceptability at successor and limit stages.
• Applying Lemma 4.2 and Lemma 4.3 to show that sets of reals constructed at successor levels are in $\widetilde{\Sigma}_\omega(\mathcal{M}^\gamma)$, ensuring definability.
• Employing the critical observation that $\rho^{n}_{\mathcal{M}^\gamma} \leq \kappa$ for some $n$ implies acceptability, derived from contradiction when $\rho^{n}_{\mathcal{M}^\gamma} > \kappa$.
• Using the iteration procedure stronger than standard premouse iteration to maintain acceptability across levels of the hierarchy.
• Proving that $\mathcal{M}^\gamma$ is a real mouse when $\rho^{n}_{\mathcal{M}^\gamma} < \kappa$, which enables the application of fine structural tools.

Experimental results

Research questions

• RQ1 Under what conditions does the boldface pointclass $\widetilde{\Sigma}_m(\mathcal{M})$ have the scale property for an iterable real premouse $\mathcal{M}$?
• RQ2 What is the minimal complexity of scales in $K(\mathbb{R})$, and how can they be constructed using fine structure?
• RQ3 Does the acceptability above the reals hold for all iterable real premice, and how does it support scale existence?
• RQ4 How does the presence of the set of reals $\mathbb{R}$ affect the fine structure and definability in $K(\mathbb{R})$?

Key findings

• Every iterable real premouse $\mathcal{M}$ is acceptable above the reals, as proven in Theorem 4.1.
• If $A \in M^{\gamma+1} \setminus M^\gamma$ is a set of reals, then $A \in \widetilde{\Sigma}_\omega(\mathcal{M}^\gamma)$, ensuring definability at the $\omega$-level.
• The proof of acceptability relies on showing $\rho^{n}_{\mathcal{M}^\gamma} \leq \kappa$ for some $n$, which prevents $M^{\gamma+1}$ from adding new reals beyond $\widetilde{\Sigma}_\omega(\mathcal{M}^\gamma)$.
• When $H^{\mathcal{M}^\gamma}_\kappa \neq H^{\mathcal{M}^{\gamma+1}}_\kappa$, it follows that $\mathcal{M}^\gamma$ is a real mouse and $\rho^{n+1}_{\mathcal{M}^\gamma} < \kappa$, which supports fine structural control.
• The acceptability result holds even when $\mathcal{M}$ does not contain all reals, as the definition of acceptability above the reals does not require $\mathbb{R}^\mathcal{M} = \mathbb{R}$.
• The result is foundational for Part II, where it is used to prove that $\widetilde{\Sigma}_m(\mathcal{M})$ has the scale property when $m = m(\mathcal{M})$ for weak real mice.