arXiv subjects
Publications and source records attributed to Farmer Schlutzenberg.
Let $λ$ be a limit of Woodin cardinals. It was shown by the second author that the pointclass of ${<λ}$-homogeneously Suslin sets has the scale property. We give a new proof of this fact, which avoids the use of stationary tower forcing.
We describe the extension of normal iteration strategies with appropriate condensation properties to strategies for stacks of normal trees, with full normalization. Given a regular uncountable cardinal $Ω$ and an $(m,Ω+1)$-iteration strategy $Σ$ for a premouse $M$, such that $Σ$ and $M$ both have appropriate condensation properties, we extend $Σ$ to a strategy $Σ^*$ for the optimal-$(m,Ω,Ω+1)^*$-iteration game such that for all $λ<Ω$ and all stacks $\vec{\mathcal{T}}=\left<\mathcal{T}_α\right>_{α<λ}$ via $Σ^*$, consisting of normal trees $\mathcal{T}_α$, each of length ${<Ω}$, there is a corresponding normal tree $\mathcal{X}$ via $Σ$ with $M^{\vec{\mathcal{T}}}_\infty=M^{\mathcal{X}}_\infty$. Moreover, if there are no drops in model or degree along the main branches of these trees then the overall iteration maps $i^{\vec{\mathcal{T}}}:M\to M^{\vec{\mathcal{T}}}_\infty$ and $i^{\mathcal{X}}:M\to M^{\mathcal{X}}_\infty$ agree. The construction is the result of a combination of work of John Steel and of the author. We also establish some further useful properties of $Σ^*$, and use the methods to analyze the comparison of multiple iterates via a common such strategy.
According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $λ$ and non-trivial elementary embedding $j:V_{λ+2}\to V_{λ+2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone has been discovered. $I_{0,λ}$ is the assertion, introduced by W. Hugh Woodin, that $λ$ is an ordinal and there is an elementary embedding $j:L(V_{λ+1})\to L(V_{λ+1})$ with critical point ${<λ}$. And $I_0$ asserts that $I_{0,λ}$ holds for some $λ$. The axiom $I_0$ is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe $V$ (in which case $λ$ must be a limit ordinal), but we assume only ZF. We prove, assuming ZF + $I_{0,λ}$ + "$λ$ is an even ordinal", that there is a proper class transitive inner model $M$ containing $V_{λ+1}$ and satisfying ZF + $I_{0,λ}$ + "there is an elementary embedding $k:V_{λ+2}\to V_{λ+2}$"; in fact we will have $k\subseteq j$, where $j$ witnesses $I_{0,λ}$ in $M$. This result was first proved by the author under the added assumption that $V_{λ+1}^\#$ exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also $λ$ is a limit ordinal and $λ$-DC holds in $V$, then the model $M$ will also satisfy $λ$-DC. We show that ZFC + "$λ$ is even" + $I_{0,λ}$ implies $A^\#$ exists for every $A\in V_{λ+1}$, but if consistent, this theory does not imply $V_{λ+1}^\#$ exists.
Suppose there is a Reinhardt cardinal. Then (1) $M_n(X)$ exists and is fully iterable (above $X$) for every transitive set $X$ and every $n<ω$ (here $M_n(X)$ denotes the canonical minimal proper class inner model containing $X$ and having $n$ Woodin cardinals above the rank of $X$); and (2) Projective Determinacy holds in every set generic extension.
We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if $W$ and $R$ are $Ω+1$-iterable, $1$-small weasels, then $W\leq^{*}R$ iff there is a club $C\subsetΩ$ such that for all $α\in C$, if $α$ is regular, then the cardinal successor of $α$ in $W$ is less or equal than the cardinal successor of $α$ in $R$ . We will show that the conjecture fails, assuming that there is an iterable premouse which models $KP$ and which has a $Σ_{1}$-Woodin cardinal. On the other hand, we show that assuming there is no transitive model of $KP$ with a Woodin cardinal the conjecture holds. In the course of this we will also show that if $M$ is an iterable admissible premouse with a largest, regular, uncountable cardinal $δ$, and $\mathbb{P}$ is a forcing poset with the $δ$-c.c. in $M$, and $g$ is $M$-generic, but not necessarily $Σ_{1}$-generic, $M[g]$ is a model of $KP$. Moreover, if $M$ is such a mouse and $T$ is maximal normal iteration tree on $M$ such that $T$ is non-dropping on its main branch, then $M_{\infty}^{T}$ is again an iterable admissible premouse with a largest regular and uncountable cardinal. At last we answer another open question from 'The Core Model Iterability Problem' regarding the S-hull property.
We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.
Let $M$ be a $λ$-indexed (that is, Jensen indexed) premouse. We prove that $M$ is iterable with respect to standard $λ$-iteration rules iff $M$ is iterable with respect to a natural version of Mitchell-Steel iteration rules. Using this equivalence, we describe a background construction for $λ$-indexed mice, analogous to traditional background constructions for Mitchell-Steel indexed mice, and which absorbs Woodin cardinals from the background universe. We also prove some facts regarding the correspondence between standard iteration trees and u-iteration trees on premice with Mitchell-Steel indexing.
Assume ZFC. Let $κ$ be a cardinal. A ${<κ}$-ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<κ}$. The $κ$-mantle is the intersection of all ${<κ}$-grounds. We prove that certain partial choice principles in the $κ$-mantle are the consequence of $κ$ being inaccessible/weakly compact, and some other related facts.
Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_δ$ be a non-trivial $\in$-cofinal $Σ_1$-elementary embedding, where $\varepsilon,δ$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_δ$, mostly focusing on the case $\varepsilon=δ$. In particular, if $\varepsilon=δ$ and $j\in L(V_δ)$ then $δ$ has cofinality $ω$. However, assuming ZFC+I$_3$, with the appropriate $\varepsilon=δ$, one can force to get such $j\in L(V^{V[G]}_δ)$. Assuming Dependent Choice and that $δ$ has cofinality $ω$ (but not assuming $V=L(V_δ)$), and $j:V_δ\to V_δ$ is $Σ_1$-elementary, we show that there are "perfectly many" such $j$, with none being "isolated". Assuming a proper class of weak Lowenheim-Skolem cardinals, we also give a first-order characterization of critical points of embeddings $j:V\to M$ with $M$ transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).
We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_α$ of the cumulative hierarchy are defined via iterated application of the power set operation, starting from $V_0=\emptyset$, and taking unions at limit stages. Assuming that $j:V_{α+1}\to V_{α+1}$ is a (non-trivial) elementary embedding, we show that the structure of $V_α$ is fundamentally different to that of $V_{α+1}$. We show that $j$ is definable from parameters over $V_{α+1}$ iff $α+1$ is an odd ordinal. Moreover, if $α+1$ is odd then $j$ is definable over $V_{α+1}$ from the parameter $j`` V_α=\{j(x)\bigm|x\in V_α\}$, and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter which is an element of $V_α$. In the case that $α=β+1$, we also give a characterization of such $j$ in terms of ultrapower maps via certain ultrafilters. Assuming $λ$ is a limit ordinal, we prove that if $j:V_λ\to V_λ$ is $Σ_1$-elementary, then $j$ is not definable over $V_λ$ from parameters, and if $β<λ$ and $j:V_β\to V_λ$ is fully elementary and $\in$-cofinal, then $j$ is likewise not definable; note that this last result is relevant to embeddings of much lower consistency strength than rank-to-rank. If there is a Reinhardt cardinal, then for all sufficiently large ordinals $α$, there is indeed an elementary $j:V_α\to V_α$, and therefore the cumulative hierarchy is eventually periodic (with period 2).
Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the elementary embedding $j:V\to V$. We investigate the linear iterates $(N_α,j_α)$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability, including that for each infinite ordinal $α$, every set is set-generic over $N_α$, but $N_α$ is not a set-ground. Assume second order ZF. We prove that the existence of super Reinhardt cardinals and total Reinhardt cardinals is not affected by small forcing. And if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no elementary embedding $j:V[G]\to M\subseteq V$ (even allowing $M$ to be illfounded).
We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let $M=(V^M,P)$ be a countable model of second order set theory $\mathsf{ZF}_2$ (with universe $V^M$ and classes $P$) which models "$κ$ is super Reinhardt". We show that there are unboundedly many $μ<κ$ such that there is $j$ such that $(V^M,j)$ models $\mathsf{ZF}(j)+$"$μ$ is Reinhardt, as witnessed by $j$". In particular, $j\upharpoonright X\in V^M$ for all $X\in V^M$ (but we allow $j\notin P$).
We identify a premouse inner model $L[\mathbb{E}]$, such that for any coarsely iterable background universe $R$ modelling $\mathrm{ZFC}$, $L[\mathbb{E}]^R$ is a proper class premouse of $R$ inheriting all strong and Woodin cardinals from $R$. Moreover, for each $α\in\mathrm{OR}$, $L[\mathbb{E}]^R|α$ is $(ω,α)$-iterable, via iteration trees which lift to coarse iteration trees on $R$. We prove that $(k+1)$-condensation follows from $(k+1)$-solidity together with $(k,ω_1+1)$-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of $(k+1)$-condensation follows from $(k,ω_1+1)$-iterability (without the $(k+1)$-solidity hypothesis). The results depend on the theory of generalizations of bicephali, which we also develop.
Let $M$ be a fine structural mouse and let $F\in M$ be such that $M\models$``$F$ is a total extender'' and $(M||\mathrm{lh}(F),F)$ is a premouse. We show that it follows that $F\in\mathbb{E}^M$, where $\mathbb{E}^M$ is the extender sequence of $M$. We also prove generalizations of this fact. Let $M$ be a premouse with no largest cardinal and let $Σ$ be a sufficient iteration strategy for $M$. We prove that if $M$ knows enough of $Σ\upharpoonright M$ then $\mathbb{E}^M$ is definable over the universe $\lfloor M\rfloor$ of $M$, so if also $\lfloor M\rfloor\models\mathrm{ZFC}$ then $\lfloor M\rfloor\models$``$V=\mathrm{HOD}$''. We show that this result applies in particular to $M=M_{\mathrm{nt}}|λ$, where $M_{\mathrm{nt}}$ is the least non-tame mouse and $λ$ is any limit cardinal of $M_{\mathrm{nt}}$. We also show that there is no iterable bicephalus $(N,E,F)$ for which $E$ is type $2$ and $F$ is type $1$ or $3$. As a corollary, we deduce a uniqueness property for maximal $L[\mathbb{E}]$ constructions computed in iterable background universes.
Let $R$ be an iterable weak coarse premouse and let $N$ be a premouse with Mitchell-Steel indexing, produced by a fully backgrounded $L[\mathbb{E}]$-construction of $R$. We identify and correct a problem with the process of resurrection used in the proof of iterability of $N$.
We develop a general theory of strategic mice, prove their condensation properties, and analyze the scales pattern in the stack of $Θ$-g-organized $\mathcal{F}$-mice over $\mathbb{R}$, Lp$^{G\mathcal{F}}(\mathbb{R})$, for a class of nice operators $\mathcal{F}$.
Assume $\mathsf{ZF}+\mathsf{AD}+V=L(\mathbb{R})$ and let $κ<Θ$ be an uncountable cardinal. We show that $κ$ is Jónsson, and that if $\mathrm{cof}(κ)=ω$ then $κ$ is Rowbottom. We also establish some other partition properties.
We describe an obstacle to the analysis of $\mathrm{HOD}^{L[x]}$ as a core model: Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are premice $M,N$ in $\mathrm{HC}^{L[x]}$ such that the pseudo-comparison of $L[M]$ with $L[N]$ succeeds, is computed in $L[x]$, and lasts through $ω_1^{L[x]}$ stages. Moreover, we can take $M=M_1|(δ^+)^{M_1}$ where $M_1$ is the minimal iterable proper class inner model with a Woodin cardinal, and $δ$ is that Woodin. We can take $N$ such that $L[N]$ is $M_1$-like and short-tree-iterable.