Iterability for (transfinite) stacks

TL;DR

This work develops a condensation-based framework (inflation condensation) that transfers normal iterability of premice to iterability for stacks of normal iteration trees. By constructing a stacks strategy $\Sigma^{\mathrm{st}}$ from a normal strategy $\Sigma$ with inflation condensation, the authors prove $(m,\Omega,\Omega+1)^*$-iterability for countable premice, via a normal realization that reduces stacks to normal trees. They establish closer connections between condensation notions (inflation, strong hull) and DJ-type properties, show how extensions to generic extensions preserve iterability and condensation, and provide absoluteness results (including universally Baire representations) under $\Omega$-cc forcing. The factor-tree viewpoint and inflation-commutativity lemmas underpin a robust theory enabling stacks to be integrated into standard iteration machinery, with explicit constructions for stacks of finite and infinite length and partial strategies. Overall, the paper advances the fine-structural toolkit for inner model theory by linking normal iterability to stack-iterability under concrete condensation hypotheses and by clarifying the generic-absolute behavior of iteration strategies.

Abstract

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $Ω$ be a regular uncountable cardinal. Let $m<ω$ and $M$ be an $m$-sound premouse and $Σ$ be an $(m,Ω+1)$-iteration strategy for $M$ (roughly, a normal $(Ω+1)$-strategy). We define a natural condensation property for iteration strategies, "inflation condensation". We show that if $Σ$ has inflation condensation then $M$ is $(m,Ω,Ω+1)^*$-iterable (roughly, $M$ is iterable for length $\leqΩ$ stacks of normal trees each of length ${<Ω}$), and moreover, we define a specific such strategy $Σ^{\mathrm{st}}$ and a reduction of stacks via $Σ^{\mathrm{st}}$ to normal trees via $Σ$. If $Σ$ has the Dodd-Jensen property and $\mathrm{card}(M)<Ω$ then $Σ$ has inflation condensation. We also apply some of the techniques developed to prove that if $Σ$ has strong hull condensation (introduced independently by John Steel) and $G$ is $V$-generic for an $Ω$-cc forcing, then $Σ$ extends to an $(m,Ω+1)$-strategy $Σ^+$ for $M$ with strong hull condensation, in the sense of $V[G]$. Moreover, this extension is unique. We deduce that if $G$ is $V$-generic for a ccc forcing then $V$ and $V[G]$ have the same $ω$-sound, $(ω,Ω+1)$-iterable premice which project to $ω$.

Figures (7)

• Figure 4: A typical picture for the embedding $\pi_{\beta\kappa}:M_{\beta\kappa}\to P_{\beta\kappa}$, when $i=i_{\beta\kappa}$ and $\gamma_{\beta i}<\gamma=\gamma_{\beta\kappa}<\delta_{\beta i}$. Note $\pi_{\beta\kappa}=j\circ\pi_{\beta i}$, where $j=j^\mathcal{X}_{\gamma_{\beta i},\gamma_{\beta\kappa}}$. The long dotted path indicates the trajectory of $\kappa$. Critical points are indicated by dotted half-headed arrows. (Where critical points are shown strictly below the image of $\kappa$ in the figure, they could in general equal that image.) Also, $\alpha\in(\gamma_{\beta i},\gamma_{\beta\kappa})_\mathcal{X}$ and $j'=i^\mathcal{X}_{\gamma_{\beta\kappa},\delta_{\beta i}}$.
• Figure 5: Commutativity of inflation, coarse and fine. In both diagrams, $\alpha_2\in C^{02}$, $\alpha_1=f^{12}(\alpha_2)$, $\alpha_0=f^{02}(\alpha_2)=f^{01}(\alpha_1)$, $\bar{\gamma}=\gamma^{01}_{\alpha_1;\alpha_0}$, $\gamma=\gamma^{02}_{\alpha_2;\alpha_0}=\gamma^{12}_{\alpha_2;\bar{\gamma}}$ and $\hat{\gamma}=\gamma^{12}_{\alpha_2;\alpha_1}$. Note $\alpha_2=\delta^{02}_{\alpha_2;\alpha_0}=\delta^{12}_{\alpha_2;\alpha_1}$ and $\alpha_1=\delta^{01}_{\alpha_1;\alpha_0}$ and $\bar{\gamma}\leq^{\mathcal{X}_1}\alpha_1$ and $\gamma\leq^{\mathcal{X}_2}\hat{\gamma}\leq^{\mathcal{X}_2}\alpha_2$. Solid arrows indicate total embeddings, and dotted arrows indicate partial embeddings (the domain and codomain are initial segments of the models in the figure). The vertical arrows are (partial) iteration embeddings. Both diagrams commute, after restricting to common domains in the fine diagram. For example, $\mathrm{dom}(\omega^{12}_{\alpha_2;\alpha_1}\circ\omega^{01}_{\alpha_1;\alpha_0})\subseteq\mathrm{dom}(\omega^ {02}_{\alpha_2;\alpha_0})$ and these maps agree over the smaller domain. Note that in the fine diagram, while the maps $\omega^{k\ell}_{\alpha_\ell;\alpha_k}$ are the only ones displayed mapping directly between segments of $M^{\mathcal{X}_k}_{\alpha_k}$ and $M^{\mathcal{X}_\ell}_{\alpha_\ell}$, there could be maps $\tau^{k\ell}_{\alpha_\ell;\alpha_k i}$ mapping between larger segments thereof, and these also commute with the rest of the diagram.
• Figure 6: Commutativity for maps relating to $\Upsilon^\Sigma_\mathcal{T}$ assuming $[0,\beta]_\mathcal{U}\cap\mathscr{D}^\mathcal{U}=\emptyset$ (see conditions \ref{['item:interval_not_easy']} and \ref{['item:interval_easy_no_mod_drop']}). The curved lines represent the iteration trees $\mathcal{T}$, $\mathcal{X}^\alpha$, $\mathcal{X}^\beta$. The solid arrows commute. The dashed arrows exist iff $b^\mathcal{T}\cap\mathscr{D}^\mathcal{T}=\emptyset$, and when they exist, they commute with the other maps.
• Figure 7: The diagram commutes, in Subcase \ref{['scase:stack_2_main']}.
• Figure 8: Commutative diagram for an infinite stack. Note that $\mathcal{U}_\beta,\varsigma^0_{\beta+1},\varsigma^1_{\beta+1}$ are not mentioned in conditions \ref{['item:lim_stack_N_0']}--\ref{['item:inflation_iteration_commutativity']}. Note that $\mathcal{Y}_0$ is trivial, $O_2=M^{\mathcal{T}_1}_\infty$ and $N_{\varepsilon+1}=M^{\mathcal{T}_\varepsilon}_\infty$. The squiggly arrows indicate inflations $\mathcal{Y}\rightsquigarrow\mathcal{Z}$; a dashed squiggly arrow indicates that $\mathcal{Z}$ is possibly $\mathcal{Y}$-terminally-model-dropping, whereas a solid squiggly arrow indicates that $\mathcal{Z}$ is $\mathcal{Y}$-terminally-non-model-dropping. The solid horizontal arrows are iteration maps; $i_{\beta\gamma}=i^{{\vec{\mathcal{T}}}\!\upharpoonright\![\beta,\gamma)}$ (and $b^{{\vec{\mathcal{T}}}\!\upharpoonright\![\beta,\gamma)}$ does not drop in model where they appear in the diagram) $i_\varepsilon=i_{\varepsilon,\varepsilon+1}$ and $j_\varepsilon=i^{\mathcal{U}_\varepsilon}$. Dotted horizontal arrows represent iteration trees possibly dropping on their main branches. Solid diagonal arrows are final inflation copy maps $\pi_\infty^{\mathcal{Y}\rightsquigarrow\mathcal{Z}}$ (such exist where they appear in the diagram). Dotted diagonal arrows represent inflations $\mathcal{Y}\rightsquigarrow\mathcal{Z}$ which are possibly $\mathcal{Y}$-terminally-model-dropping. Vertical arrows are the lifting maps $\varsigma_\delta$. All solid arrows commute.

Theorems & Definitions (217)

• Theorem : \ref{['thm:stacks_iterability']}, \ref{['thm:stacks_iterability_2']}
• Definition 1.1
• Theorem : \ref{['thm:stacks_of_finite_trees']}
• Theorem 1.2
• proof
• Theorem : \ref{['thm:strat_with_cond_extends_to_generic_ext']}
• Corollary : \ref{['cor:wDJ_absoluteness']}, \ref{['cor:stack_it_absoluteness']}
• Theorem : \ref{['thm:Sigma_shc_implies_stacks_npc']}
• Remark 2.1
• Definition 2.2