The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals

TL;DR

The paper studies how the failure of the square principle at all uncountable cardinals can be realized in a ZFC-extension arising from forcing over a minimal Nairian model, and how this interacts with strong HOD structure. Using a ${P_{ m max}}$-style forcing followed by a full-support iteration, the authors force AC while preserving as much determinacy structure as possible, obtaining a model in which ${\square}(\gamma)$ fails for all uncountable $\gamma$ (with certain cofinality constraints) and every regular cardinal is ${\omega}$-strongly measurable in ${\rm HOD}$, witnessed by the $\,\omega$-club ultrafilter $F_{\kappa}({\rm HOD})$. This yields a nuanced view of the relative consistency strength of square failures, PFA-type consequences, and the HOD hypothesis, showing that the HOD hypothesis is not provable in ZFC and highlighting limits to inner-model-based lower bounds via square failures. The work connects high-level determinacy-driven inner model theory (including Hod Pair Capturing and minimal Nairian models) with forcing techniques to produce a model where square fails broadly yet regular cardinals exhibit strong HOD-measurability, informing ongoing discussions about the interplay between large cardinals, square principles, and forcing axioms. Overall, the results illuminate a regime where failures of square can be forced weakly relative to Woodin-like hierarchies, while maintaining a robust HOD structure and challenging existing conjectures about provability of the HOD hypothesis in ZFC.

Abstract

We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC.

Figures (35)

• Figure 10.1: Corollary \ref{['cor: technical corollary']}
• Figure 10.2: The iterations leading from ${\mathfrak{s}}'$ and ${\mathfrak{r}}'$ to ${\mathfrak{w}}'$.
• Figure 10.3: The relationship between ${\mathfrak{q}}$, ${\mathfrak{q}}'$, ${\mathfrak{s}}'$, and ${\mathfrak{w}}'$.
• Figure 10.4: The iteration $\mathcal{T}_{{\mathfrak{s}}, {\mathfrak{w}}'}$ projecting towards ${\mathfrak{w}}$, and the subsequent agreement between ${\mathfrak{w}}$ and ${\mathfrak{w}}'$.
• Figure 10.5: The contradiction in Claim \ref{['clm: crucial']}: If the first strong cardinals differ ($\tau_{\mathfrak{x}} > \tau_{\mathfrak{y}}$), ${\mathfrak{y}}$ becomes a proper segment of ${\mathfrak{x}}$, forcing an extender usage that was previously ruled out.

Theorems & Definitions (200)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Todorčević To84
• Corollary 1.5
• Theorem 1.6: Neeman-Steel ESC
• Conjecture 1.8
• Corollary 1.9
• Theorem 1.10
• Theorem 1.11