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Publications and source records attributed to Farmer Schlutzenberg.
Let $M$ be a $(0,ω_1+1)$-iterable mouse with no largest cardinal. Let $γ\leqκ$ be uncountable cardinals of $M$, with $κ$ regular in $M$. Then $M$ satisfies $\diamondsuit_{κγ}^+$, and if $κ$ is non-ineffable in $M$ then $M$ satisfies $\diamondsuit_{κκ}^+$. Suppose that either $ω_1^M=ω_1$ or $M$ satisfies "I am $(0,ω_1+1)$-iterable". Let $κ\geqω_1^M$ be a cardinal of $M$. Then $M$ satisfies $\diamondsuit^*_{κω_1^M}$.
We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $κ$ is a limit of measurables then there is no wellorder of a subset of $P(κ)$ of length $\geqκ^+$ which is $Σ_1(V_κ\cup\mathrm{OR})$, answering a question of Lücke and Müller. However, consistently, a Woodin cardinal exists and for every uncountable cardinal $κ$ which is not a limit of measurables, there is a $Σ_1(H_κ\cup\{κ\})$-good wellorder of $H_{κ^+}$. If $κ$ is a limit of measurables and $κ$ has uncountable cofinality then there is no $Σ_1(V_κ\cup\mathrm{OR})$ almost disjoint family $F\subseteq P(κ)$ of cardinality $>κ$. Consistently, $Π_1(\{κ\})$ mad families and maximal independent families $F\subseteq P(κ)$ exist, $κ$ is a limit of measurables, and more. If $κ$ is weakly compact and every $Σ_1(V_κ\cup\{κ\})$ subset of $P(κ)$ of cardinality $>κ$ contains a perfect subset of the right kind, then there is an inner model with a weakly compact limit of measurables. We prove some related facts regarding $Σ_1(V_λ\cup\{V_λ\}\cup\mathrm{OR})$ when $I_2(λ)$ holds. These depend on an analysis of fixed points of linear iterations involving $I_2(λ)$-extenders.
Väänänen and Welch asked in the paper "When cardinals determine the power set: inner models and Härtig quantifier logic" which large cardinals are consistent with the power set operation $x\mapsto P(x)$ being $Σ_1$-definable in the predicate Card of all cardinals. We show that, relative to large cardinals, this property is consistent together with the existence of two Woodin cardinals.
The technique of $\star$-translation is important in arguments calibrating the strengths of determinacy theories against large cardinals, for example in [9] and [1] in the paper's bibliography. It has also been used in analysing the internal theory of mice, for example in [5], [3], [6]. We give a detailed development of $\star$-translation, slightly strengthening the large cardinal level in the development of [1]. We then develop a variant of $\star$-translation for Varsovian models, in which certain extenders overlapping the Woodin cardinal are total, and used to directly induce extenders on the sequence of the $\star$-translation (details for this component are missing from this draft). This variant was used in [3] and [6], where it was outlined but not developed in detail. We use the material to verify the star-translation hypothesis of [5] and deduce some self-iterability facts.
We give a development of the fine structure of mice with long extenders, to the level of $κ^+$-supercompact cardinals $κ$. We do this using a hierarchy with features more analogous to those familiar in the short extender context than the hierarchies introduced by Woodin and by Neeman-Steel. In particular, the mice we consider satisfy stronger versions of the initial segment condition. We establish a form of fine structural condensation involving embeddings $π:H\to M$ which need not be the identity below the projectum of $H$ (under special assumptions). We also adapt the analysis of the Dodd structure of short extenders on the sequence to mice at this level.
We extend the normalization results of the author's paper "Full normalization for transfinite stacks" [5] to mice at the level of $κ^+$-supercompactness: given a normal iteration strategy $Σ$ for such a mouse $M$, with both $M$ and $Σ$ satisfying certain condensation properties, we extend $Σ$ to a strategy $Σ^*$ for stacks of normal trees, such that every iterate via $Σ^*$ is in fact a normal iterate via $Σ$.
We show that (i) the standard fine structural properties for premice follow from normal iterability (whereas the classical proof relies on iterability for stacks of normal trees), and (ii) every mouse which is finitely generated above its projectum, is an iterate of its core. That is, let $m$ be an integer and let $M$ be an $m$-sound, $(m,ω_1+1)$-iterable premouse. Then (i) $M$ is $(m+1)$-solid and $(m+1)$-universal, $(m+1)$ condensation holds for $M$, and if $m\geq 1$ then $M$ is super-Dodd-sound, a slight strengthening of Dodd-soundness. And (ii) if there is $x\in M$ such that $M$ is the $\mathrm{r}Σ_{m+1}$-hull of parameters in $ρ_{m+1}^M\cup\{x\}$, then $M$ is a normal iterate of its $(m+1)$-core $C=\mathfrak{C}_{m+1}(M)$; in fact, there is an $m$-maximal iteration tree $\mathcal{T}$ on $C$, of finite length, such that $M=M^{\mathcal{T}}_\infty$, and $i^{\mathcal{T}}_{0\infty}$ is just the core embedding. Applying fact (ii), we prove that if $M\models\mathrm{ZFC}$ is a mouse and $W\subseteq M$ is a ground of $M$ via a strategically $σ$-closed forcing $\mathbb{P}\in W$, and if $M|\aleph_1^M\in W$ (that is, the initial segment of $M$ of height $\aleph_1^M$ is in $W$), then the forcing is trivial; that is, $M\subseteq W$. And if there is a measurable cardinal, then there is a non-solid premouse. The results hold for premice with Mitchell-Steel indexing, allowing extenders of superstrong type to appear on the extender sequence.
Assume the existence of sufficent large cardinals. Let $M_{\mathrm{sw}n}$ be the minimal iterable proper class $L[E]$ model satisfying "there are $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that the $δ_i$ are Woodin cardinals and the $κ_i$ are strong cardinals". Let $M=M_{\mathrm{sw}2}$. We identify an inner model $\mathscr{V}_2^M$ of $M$, which is a proper class model satisfying "there are 2 Woodin cardinals", and is iterable both in $V$ and in $M$, and closed under its own iteration strategy. The construction also yields significant information about the extent to which $M$ knows its own iteration strategy. We characterize the universe of $\mathscr{V}_2^M$ as the mantle and the least ground of $M$, and as $\mathrm{HOD}^{M[G]}$ for $G\subseteq\mathrm{Coll}(ω,λ)$ being $M$-generic with $λ$ sufficiently large. These results correspond to facts already known for $M_{\mathrm{sw}1}$, and the proofs are an elaboration of those, but there are substantial new issues and new methods used to handle them.
For $n<ω$, let $N_n$ be the minimal iterable proper class mouse $M$ such that $M\models$ "there are ordinals $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that each $δ_i$ is a Woodin cardinal and each $κ_i$ is a strong cardinal", and let $N_ω$ be likewise, but with $M\models$ "there is an ordinal $λ$ which is a limit of Woodin cardinals and a limit of strong cardinals". Under appropriate large cardinal hypotheses, Sargsyan and Schindler introduced and analysed in "Varsovian models I" the Varsovian model of $N_1$, and Sargsyan, Schindler and the author introduced and analysed in "Varsovian models II" the Varsovian model of $N_2$. We extend this to $N_ω$, assuming that $*$-translation integrates routinely with the P-constructions of this paper (the write-up of which is yet to be completed). We show, under this assumption, that $N_ω$ has a proper class inner model $\mathscr{V}_ω$ which is a fully iterable strategy mouse with $ω$ Woodin cardinals, closed under its strategy, and that the universe of $\mathscr{V}_ω$ is the eventual generic HOD, and the mantle, of $N_ω$. We also show, under the same assumption, that the core model $K$ of $N_ω$ (which can be defined in a natural manner) is an iterate of $N_ω$, is an inner model of $\mathscr{V}_ω$, and is fully iterable in $M$ and in $\mathscr{V}_ω$.
Assume ZF + AD + V=L(R). Let $[α,β]$ be a $Σ_1$ gap with $J_α(R)$ admissible. We analyze $J_β(R)$ as a natural form of "derived model" of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_β(R)=\mathcal{P}(R)\cap D$, and if $J_β(R)\models$ "$Θ$ exists", then $J_β(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R))^M$, for $ω$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper.
We give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is related to the Martin-Solovay construction for scales on $Π^1_2$ sets. The prewellorders of the scales compare reals $x$ and $y$ by comparing features of certain kinds of fully backgrounded $L[E,x]$- and $L[E,y]$-constructions executed in mice $P$ with $x,y \in P$. In this way we produce an inner model theoretic proof of the scale property for many pointclasses, for which the scale property was classically established using determinacy arguments (for example, $Π^1_3$). Moreover, it also yields many further pointclasses with the scale property, for example intermediate between $Π^1_{2n+1}$ and $Σ^1_{2n+2}$, and also instances of complexity well beyond projective.
Let $M$ be a tame mouse modelling ZFC. We show that $M$ satisfies "$V=\mathrm{HOD}_x$ for some real $x$", and that the restriction $\mathbb{E}\upharpoonright[ω_1^M,\mathrm{OR}^M)$ of the extender sequence $\mathbb{E}^M$ of $M$ to indices above $ω_1^M$ is definable without parameters over the universe of $M$. We show that $M$ has universe $\mathrm{HOD}^M[X]$, where $X=M|ω_1^M$ is the initial segment of $M$ of height $ω_1^M$ (including $\mathbb{E}^M\upharpoonrightω_1^M$), and that $\mathrm{HOD}^M$ is the universe of a premouse over some $t\subseteqω_2^M$. We also show that $M$ has no proper grounds via strategically $σ$-closed forcings. We then extend some of these results partially to non-tame mice, including a proof that many natural $φ$-minimal mice model "$V=\mathrm{HOD}$", assuming a certain fine structural hypothesis whose proof has almost been given elsewhere.
We develop the fine structure theory of operator-premice. These are a generalization of standard premice, in which an abstract operator $F$ is used to form the successor steps in the internal hierarchy of the premouse, instead of Jensen's $J$-operator (which computes rudimentary closure). Such notions have seen applications in core model induction arguments, but their theory has not previously been developed in detail. We define fine condensation for operators $F$ and show that fine condensation and iterability together ensure that $F$-mice have the fundamental fine structural properties including universality and solidity of the standard parameter.
Assume ZFC. Let $κ$ be a cardinal. Recall that a ${<κ}$-ground is a transitive proper class $W$ modelling ZFC such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<κ}$, and the $κ$-mantle is the intersection of all ${<κ}$-grounds. Assume there is a Woodin cardinal and a proper class of measurables, and let $x$ be a real of sufficiently high Turing degree. Let $κ$ be a limit cardinal of $L[x]$ of uncountable cofinality in $L[x]$. Using methods from Woodin's analysis of $\mathrm{HOD}^{L[x,G]}$, we analyze the $κ$-mantle of $L[x]$, and show that it models ZFC + GCH + "There is a Woodin cardinal". Moreover, we show that it is a fully iterable strategy mouse (analogous to $\mathrm{HOD}^{L[x,G]}$). We also analyze another form of "local mantle", partly assuming also a weak form of Turing determinacy. We also compute bounds on how much iteration strategy can be added to $M_1$ before $M_1^\#$ is added.
Let $M$ be a short extender mouse. We prove that if $E\in M$ and $M$ satisfies "$E$ is a countably complete short extender whose support is a cardinal $θ$ and $\mathcal{H}_θ\subseteq\mathrm{Ult}(V,E)$", then $E$ is in the extender sequence $\mathbb{E}^M$ of $M$. We also prove other related facts, and use them to establish that if $κ$ is an uncountable cardinal of $M$ and $κ^{+M}$ exists in $M$ then $(\mathcal{H}_{κ^+})^M$ satisfies the Axiom of Global Choice. We prove that if $M$ satisfies the Power Set Axiom then $\mathbb{E}^M$ is definable over the universe of $M$ from the parameter $X=\mathbb{E}^M\upharpoonright\aleph_1^M$, and $M$ satisfies "every set is $\mathrm{OD}_{\{X\}}$". We also prove various local versions of this fact in which $M$ has a largest cardinal, and a version for generic extensions of $M$. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models "$V=\mathrm{HOD}$". This adapts to many other similar examples. We also describe a simplified approach to Mitchell-Steel fine structure, which does away with the parameters $u_n$.
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 $ω$.
Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \mathbb{R}$ let $α_x$ be the least $β$ such that \[ L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is inaccessible and }κ^+ \text{ exists}"). \] We adapt the analysis of $\text{HOD}^{L[x,G]}$ as a strategy mouse to $L_{α_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse $\mathcal{M}^{\text{n-ad}}$ and define a class $H \subseteq L_{α_x}[x,G]$ as a natural analogue of $\text{HOD}^{L[x,G]} \subseteq L[x,G]$, and show that $H = M_\infty[Σ_0]$, where $M_\infty$ is an iterate of $\mathcal{M}^{\text{n-ad}}$ and $Σ_0$ a fragment of its iteration strategy.
Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_α(\mathbb{R})$ where $[α,α]$ is a projective-like gap (of $L(\mathbb{R})$) and $α$ is either a successor ordinal or has countable cofinality, but $α\neqβ+1$ where $β$ ends a strong gap. For such ordinals $α$ and integers $n\geq 1$, we show that there is a mouse $M$ with $\mathbb{R}\cap M=\mathrm{OD}_{αn}$. The proof involves an analysis of ladder mice and their generalizations to $J_α(\mathbb{R})$. This analysis is related to earlier work of Rudominer, Woodin and Steel on ladder mice. However, it also yields a new proof of the mouse set theorem even at the least point where ladder mice arise -- one which avoids the stationary tower. The analysis also yields a corresponding "anti-correctness" result on a cone, generalizing facts familiar in the projective hierarchy; for example, that $(Π^1_3)^V\upharpoonright M_1$ truth is $(Σ^1_3)^{M_1}$-definable and $(Σ^1_3)^{M_1}$ truth is $(Π^1_3)^V\upharpoonright M_1$-definable. We also define and study versions of ladder mice on a cone at the end of weak gap, and at the successor of the end of a strong gap, and an anti-correctness result on a cone there.