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Nairian Models

Douglas Blue, Paul B. Larson, Grigor Sargsyan

TL;DR

The paper introduces Nairian models—canonical Chang-type inner models built from HODs of AD-type universes—and develops forcing over them to achieve models of ZFC with MM^{++}(c) while simultaneously failing Jensen’s square principles at multiple ω_ classes. It integrates derived-model techniques, universally Baire settings, and the P_{ ext{max}} framework to force MM^{++} over determinacy contexts, and demonstrates obstructions to extending inner model methods beyond Woodin cardinals that are limits of Woodin cardinals. The authors establish a main theorem constructing Nairian models from hod mice, show that powerset computations agree across Nairian realizations, and derive corollaries including negative answers to questions about ω1-supercompactness equiconsistency and the iterability of K^c constructions. These results reveal inherent obstructions to extending descriptive inner model theory beyond certain Woodin-based barriers and illuminate a deep interplay between determinacy, forcing axioms, and inner model techniques.

Abstract

We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\neg\square_{ω_3}+\neg\square(ω_3)$. Then, fixing $n\in [3, ω)$, we design a Nairian model and force over it to produce a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\forall i\in [2, n]\, \neg\square(ω_i)$. We also build a Nairian model that satisfies ${\sf{ZF}}+"ω_1$ is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler $\sf{K}^{c}$ construction, (2) the consistent failure of the Iterability Conjecture for the $\sf{K}^{c}$ construction using $2^{2^{\dots 2^ω}}$-complete (for any finite stack of exponents) background extenders, answering a strong version of a question asked by Steel, and (3) a negative answer to Trang's question whether ${\sf{ZF}}+"ω_1$ is a supercompact cardinal" is equiconsistent with ${\sf{ZFC}}+"$there is a proper class of Woodin cardinals that are limits of Woodin cardinals." These corollaries identify obstructions to extending the methods of (descriptive) inner model theory past a Woodin cardinal which is a limit of Woodin cardinals.

Nairian Models

TL;DR

The paper introduces Nairian models—canonical Chang-type inner models built from HODs of AD-type universes—and develops forcing over them to achieve models of ZFC with MM^{++}(c) while simultaneously failing Jensen’s square principles at multiple ω_ classes. It integrates derived-model techniques, universally Baire settings, and the P_{ ext{max}} framework to force MM^{++} over determinacy contexts, and demonstrates obstructions to extending inner model methods beyond Woodin cardinals that are limits of Woodin cardinals. The authors establish a main theorem constructing Nairian models from hod mice, show that powerset computations agree across Nairian realizations, and derive corollaries including negative answers to questions about ω1-supercompactness equiconsistency and the iterability of K^c constructions. These results reveal inherent obstructions to extending descriptive inner model theory beyond certain Woodin-based barriers and illuminate a deep interplay between determinacy, forcing axioms, and inner model techniques.

Abstract

We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of . Then, fixing , we design a Nairian model and force over it to produce a model of . We also build a Nairian model that satisfies is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler construction, (2) the consistent failure of the Iterability Conjecture for the construction using -complete (for any finite stack of exponents) background extenders, answering a strong version of a question asked by Steel, and (3) a negative answer to Trang's question whether is a supercompact cardinal" is equiconsistent with there is a proper class of Woodin cardinals that are limits of Woodin cardinals." These corollaries identify obstructions to extending the methods of (descriptive) inner model theory past a Woodin cardinal which is a limit of Woodin cardinals.

Paper Structure

This paper contains 48 sections, 71 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Core model theory and the iterability problem for $\sf{K^c}$
  3. Forcing over models of determinacy
  4. Higher models of determinacy
  5. Universally Baire sets and Chang models over derived models.
  6. Future research
  7. The Nairian cardinals
  8. Preliminaries
  9. Square principles
  10. The Nairian Model at $\lambda$ and $\Join^\xi_{\lambda}$
  11. Variants of $\mathsf{DC}$
  12. $\mathbb{P}_{\mathrm{max}}$
  13. Forcing over Nairian models
  14. Threading coherent sequences
  15. Proof of Theorem \ref{['mainthrm']}
  16. ...and 33 more sections

Key Result

Theorem 1.2

Assume $\aleph_2^\omega=\aleph_2$, and suppose that the principles $\square(\omega_3)$ and $\square_{\omega_3}$ both fail. Let $g\subseteq \mathrm{Col}(\omega_3, \omega_3)$ be a $V$-generic filter. If $V[g]\vDash "{\sf{K^c_{MiSch}}}$ converges," then $({\sf{K^c_{MiSch}}})^{V[g]}\vDash$ "there is a s

Figures (2)

  • Figure 9.1: Lemma \ref{['splitting lemma']}. $\mathcal{Y}$ is strictly above $\pi_{\mathfrak{q},\mathfrak{w}}(\nu)$.
  • Figure 9.2: Lemma \ref{['bound iterations factor']}. $\mathcal{X}$ is the full normalization of $({\mathcal{T}}_{{\mathfrak{q}}, {\mathfrak{r}}})^\frown {\mathcal{T}}^{\nu_{\mathfrak{r}}}_{{\mathfrak{r}}, \infty}$, and ${\rm crit }(\pi^{\mathcal{X}}_{\xi, \zeta})>\nu_\infty$.

Theorems & Definitions (152)

  • Definition 1.1
  • Theorem 1.2: Jensen-Schimmerling-Schindler-Steel JSSS
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7: Woodin
  • Definition 1.9: Feng-Magidor-Woodin, FMW92
  • Theorem 1.11: Woodin, St07DMT
  • Theorem 1.12: Sargsyan, CCM
  • ...and 142 more