Set-Theoretic Mice: Fine-Structural Models

• Set-theoretic mice are canonical, fine-structural inner models built using extender sequences and iteration strategies to capture fragments of large cardinal strength and descriptive-set complexity.
• They exhibit key properties such as soundness, condensation, and universality, which support precise analysis of definability and determinacy phenomena in set theory.
• Recent developments like ladder mice extend the framework to encompass projective-like and admissible gaps, eliminating the need for stationary-tower forcing in canonical constructions.

A set-theoretic mouse is a canonical, fine-structural, typically countable, inner model equipped with an extender sequence and a normal iteration strategy, engineered to encapsulate precise fragments of large cardinal strength and descriptive-set-theoretic complexity. Mice play a central role at the interface of descriptive set theory, inner model theory, and the structural analysis of definability and determinacy, serving as the key objects in canonical inner models for large cardinals (notably Woodin cardinals) and as the organizing principle for reducibility analyses of definable sets of reals. These objects are characterized both by their fine-structural properties (soundness, condensation, hull properties) and by their interaction with iteration trees and extender sequences, yielding a hierarchically stratified universe of canonical inner models. Recent advances, such as ladder mice, push the boundaries of the mouse framework to capture definability at projective-like and admissible gaps of the $L(\mathbb{R})$ hierarchy, with profound implications for the structure of ordinal definability, anti-correctness phenomena, and the elimination of stationary-tower arguments from key theorems (Schlutzenberg, 2024).

1. Fine Structure and Formal Definition of Mice

Let $M$ be a premouse—a transitive structure $\langle J_\alpha[E], \in, E\rangle$ upholding a fine-structural hierarchy with a coherent sequence of (possibly partial, possibly overlapping) extenders $E$ indexed below $\alpha$. The fine structure comprises notions such as projecta $\rho_n^M$, iterability via normal iteration trees, Dodd parameters, and condensation/comparison properties. A mouse is a premouse $M$ equipped with a specified normal iteration strategy $\Sigma_M$ (typically countably iterable, i.e., $(0, \omega_1+1)$-iterable), guaranteeing the existence of unique wellfounded branches through all countable normal iteration trees.

The prototypical example is $M_n^\#$, the minimal sound, countable premouse with $n$ Woodin cardinals, which is $(\omega, \omega_1)$-iterable and has no total extenders above its last Woodin (Zhu, 2016, Zhu, 2017).

Core properties:

• Soundness and Universality: Mice are solid and universal at each level of fine-structural projecta. If $M$ is $m$-sound and $(m,\omega_1+1)$-iterable, both $(m+1)$-solidity and $(m+1)$-universality hold (Schlutzenberg, 2020).
• Condensation and Hulls: Each $m$-sound mouse admits condensation relative to its $(m+1)$-projectum, and, if generated as the $\mathrm{r}\Sigma_{m+1}$-hull over $\rho_{m+1}^M\cup\{x\}$, is a finite iterate of its core.
• Super-Dodd Soundness: Active, 1-sound mice that are $(0,\omega_1+1)$-iterable exhibit super-Dodd-soundness and Dodd-amenability (Schlutzenberg, 2020).

2. Extender Sequences, Measures, and Iterability

Extenders, as systems of partial ultrafilters indexed over subsets of ordinals, give rise to ultrapower embeddings and are the combinatorial engines behind large cardinal strength in mice. The canonical extender sequence $E^M$ in a mouse $M$ is uniquely determined by the fine-structural and coherence constraints (initial-segment condition, no overlap, etc.).

Uniqueness and Fine Structure:

• Kunen’s Uniqueness asserts that in mice such as $L[U]$ (the inner model for a measurable), there is exactly one normal measure, a fact that generalizes to mice below a superstrong cardinal: any wellfounded, countably complete, short extender with critical point a cardinal and premouse support appears on the extender sequence (Schlutzenberg, 2013).
• Extender-full Self-iterability: Mice such as $M_n$ possess extender-full self-iterability at their cardinals: any initial segment projecting to a cardinal $\eta$ knows its own $(\eta^++1,\eta)$-extender-full strategy, enabling the entire extender sequence, and hence the mouse, to be definable from parameters (Schlutzenberg, 2013).

Iteration Trees and Strategies:

• Any countable mouse is equipped with a unique iteration strategy (typically $(\omega, \omega_1)$-iterability is required for applications to descriptive set theory), which directs the selection of branches in normal trees of extenders.
• Ladder mice, in particular, are built by constructing sequences of cutpoints (usually Woodin cardinals) and enforcing that every proper initial segment is absorbed by lower approximations via the $\mathsf{Lp}_\Gamma$ construction over substructures (Schlutzenberg, 2024).

3. Canonical Examples and Hierarchy of Mice

Mice are classified according to the large cardinal structure of their extender sequence and the complexity of their projecta:

• $M_n^\#$ Mice: The canonical mouse with $n$ Woodin cardinals and no total extenders above, providing the inner model for many projective sets.
• Ladder Mice: Generalizations of $M_{2n-1}$, constructed to capture definability over $J_\alpha(\mathbb{R})$ at projective-like and admissible gaps, with combinatorial anti-correctness properties and Δ₂-definability (Schlutzenberg, 2024).
• Machete Mice ($\mathsf{OM}_\alpha$): Minimal active, sound, small mice characterized by Cantor–Bendixson ranks of measurable cardinals, generating $L[R]$-type models via Magidor iteration and Prikry forcings (Henney-Turner et al., 2024).

Hierarchy Table:

Mouse Type	Large Cardinal Content	Typical Complexity
$M_n^\#$	$n$ Woodin cardinals	$\Pi^1_{2n+1}$
Ladder Mouse	ω-sequence of Woodins (laddered)	Projective-like or admissible gaps
$\mathsf{OM}_\alpha$	Measurables $\mathrm{CB}$-rank < $\alpha$	Below $O^{\mathsf{Sword}}$

4. Definability, Determinacy, and Mouse Set Theorems

Mice provide canonical witnesses to definability and determinacy phenomena:

• Mouse Set Theorem: For projective-like gaps $[\alpha,\alpha]$ in $L(\mathbb{R})$ with $\operatorname{cf}(\alpha) = \omega$ and not the successor of a strong gap, there exists a countable 1-small, $(0,\omega_1+1)$-iterable premouse $M$ with $\mathbb{R}\cap M = \mathrm{OD}_{\alpha n}$ for any $n\geq 1$ (Schlutzenberg, 2024).
• Anti-correctness: For such $M$, $\Pi_2$-formulas over $J_\alpha(\mathbb{R})$ are equivalent to recursively assigned $\Sigma_2$-formulas over $M$. Conversely, $\Pi_2$ truth in $M$ is $\Sigma_2$-definable in $J_\alpha(\mathbb{R})$ (Schlutzenberg, 2024).
• Ordinal Definability and HOD: In tame mice, the universe is $\mathrm{HOD}_x$ for some real $x\in M$, and the parameter-free definability of the tail of the extender sequence is achievable (Schlutzenberg, 2020, Schlutzenberg, 2013).

Set-theoretic Implications:

• The existence of certain mice is equiconsistent with strong determinacy hypotheses; e.g., $\Delta^1_{2n+2}$-determinacy and $\boldsymbol{\Pi}^1_{2n+1}$-determinacy imply the existence of an $(\omega, \omega_1)$-iterable $M_{2n+1}^\#$ (Zhu, 2016).
• Under ZF+$\mathrm{AD}$+$V=L(\mathbb{R})$, the entire mouse set and anti-correctness program can be realized at the level of ladder mice analysis, eliminating the use of stationary-tower forcing techniques (Schlutzenberg, 2024).

5. Applications: Scales, Suslin Sets, and the Mouse Set Conjecture

Mice unify descriptive set theory and inner model theory:

• Homogeneously Suslin Sets: In $M_n$, all homogeneously Suslin sets of reals are $\mathbf{\Delta}^1_{n+1}$, and in particular, these sets are $M_n$-correctly $\Delta^1_{n+1}$, matching their universally Baire properties (Schlutzenberg, 2013).
• Mouse Set Conjecture (MSC): Under $AD^+$ and $V=L(P(\mathbb{R}))$, a real is ordinal-definable if and only if it is contained in an $\omega_1$-iterable mouse. The analogous result for sets of reals follows in strong models, with sound countably iterable $\Sigma$-mice over $\mathbb{R}$ capturing all sets $A\subseteq\mathbb{R}$ ordinal-definable from a real (Sargsyan et al., 2021).
• Axiom of Determinacy and Choice: In mice constructed over their reals and satisfying $\mathrm{AD}$, $\mathrm{DC}$ holds, reinforcing the deep correspondences between game-theoretic principles and fine-structural inner models (Müller, 2019).
• Relation to Core Models: Mice serve as stepping stones towards the construction of the ultimate core model for large cardinals, with each class providing universality at its projective or admissible gap level.

6. Eliminating Stationary-Tower Forcing and Broad Implications

One of the most significant recent advances is the replacement of stationary-tower forcing arguments with pure combinatorial analysis of the internal structure of minimal ladder and related mice. When considering definability, mouse-set theorems, and anti-correctness at or above projective and admissible gaps, the construction of iteration trees and the combinatorics of ladder rungs (cutpoints) suffice to realize canonical sets without recourse to external genericity arguments (Schlutzenberg, 2024). This brings greater uniformity and depth to the analysis of the mouse hierarchy and the structural theory of $\mathrm{OD}$ sets of reals.

Significance:

• The analysis of set-theoretic mice provides canonical bridges between descriptive set theory (e.g., scales, pointclasses) and inner model theory (extenders, iteration trees).
• Ladder mice and their generalizations extend the field of the mouse set phenomenon, embedding canonical definability at broader levels of the $L(\mathbb{R})$ hierarchy.
• These techniques show that definability, anti-correctness, and canonical inner model existence can all be internalized within fine-structural combinatorics, bypassing previous reliance on stationary towers and external forcing.

Broader Implications Table:

Development	Consequence	Reference
Ladder mice (internal analysis)	Mouse-set and anti-correctness at projective-like and admissible gaps	(Schlutzenberg, 2024)
Extender-full iterability	$V=HOD_x$ and parameter-free definability	(Schlutzenberg, 2020, Schlutzenberg, 2013)
Fine-structure from iterability	Solidity, universality, core-iteration	(Schlutzenberg, 2020)
MSC and SMSC	Canonical capture of $OD$ (and $OD(x)$) sets	(Sargsyan et al., 2021)

7. Future Directions and Open Questions

The current landscape positions set-theoretic mice as canonical fine-structural models for foundational questions in the definability and classification of sets of reals, the analysis of determinacy, and inner model theory for large cardinals. Open directions include:

• Pushing ladder-mouse constructions beyond gaps of countable cofinality or into the field of full $\omega$-Woodin mice.
• Generalizing the parameter-free definability of extender sequences to broader classes of non-tame or hybrid mice.
• Systematic unification of descriptive-set-theoretic and structural inner model phenomena at higher levels of the complexity hierarchy.

The interplay between fine structure, iteration strategies, and canonicality in mice continues to dictate both the formulation and resolution of central problems in set theory and the theory of definability.