This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega-Jonsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.
We show that the notions of generic and Laver-generic supercompactness are first-order definable in the language of ZFC. This also holds for generic and Laver-generic (almost) hugeness as well as for generic versions of other large cardinals.
Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $E$ be a $\mathbfΣ^1_1$ equivalence relation coded in $\mathrm{HOD}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\mathrm{HOD} \models E$ is unpinned. $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$ proves $E$-class section uniformization when $E$ is a $\mathbfΣ^1_1$ equivalence relation on $\mathbb{R}$ which is pinned in every transitive model of $\mathsf{ZFC}$ containing the real which codes $E$: Suppose $R$ is a relation on $\mathbb{R}$ such that each section $R_…
Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $κ$, there are no $Σ^1_1(κ)-κ-$mad families.
We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets, that is, HOA = HOD. Moreover, we show that every (pointwise) algebraic model of ZF is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp---an algebraic analogue of the constructible universe---which is obtained by iteratively adding not only the sets that are definable over wh…
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a $Δ_2$-definable class containing no regular cardinals. In such a model, there can be no $Σ_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
We will consider a number of new large-cardinal properties, the $α$-tremendous cardinals for each limit ordinal $α>0$, the hyper-tremendous cardinals, the $α$-enormous cardinals for each limit ordinal $α>0$, and the hyper-enormous cardinals. For limit ordinals $α>0$, the $α$-tremendous cardinals and hyper-tremendous cardinals have consistency strength between I3 and I2. An $ω$-enormous cardinal has consistency strength greater than I0, and also all the large-cardinal axioms discussed in the second part of Hugh Woodin's paper on suitable extender models, not known to be inconsistent with ZFC an…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and fin…
We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of $\square_κ$ in terms of local club condensation in extender models. Assuming $\gch$, given an interval of ordinals $I$ we verify that iterating the forcing defined by Holy-Welch-Wu, we can preserve $\gch$, cardinals and cofinalities and obtain a model where local club condensation holds for every ordinal in $I$ modulo those ordinals which cardinality is a si…
It is shown that the existence of a measurable cardinal is equiconsistent to a model of ZFC in which there is no ordinal-definable, stationary, costationary subset of $ω_1$
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels $V_α$ of the cumulative hierarchy of sets that are eventually periodic, alternating according to the parity of the ordinal $α$. For example, if there is an elementary embedding from the universe of sets to itself, then for sufficiently large ordinals $α$, the supremum of the lengths of all wellfounded relations on $V_α$ is a strong limit cardinal if and only if…
We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal $κ$ such that $2^κ> kappa^+$ then there is an inner model with a cardinal $κ$ such that for all ordinals $α<κ$ there is an ordinal $ν< κ$ with $o(ν) > α$. (ii) If there is a singular strong limit cardinal $κ$ of uncountable cofinality such that $2^κ> κ^+$ then there is an inner model with $o(κ) = κ^{++}$. Since this paper was originally submitted, Gitik has improved this result to give exact lower bounds.
We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a set of ordinals in $V$ cannot be added to $L({\mathbb R})$ by small forcing. The large cardinal needed corresponds to the consistency strength of $AD^{L({\mathbb R})}$; roughly $ω$ Woodin cardinals.
We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowan…
Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_δ$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_δ$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $δ$, using a ground model parameter. In particular, this means that models of…
We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{k…
If $G$ is a centreless group, then $τ(G)$ denotes the height of the automorphism tower of $G$. We prove that it is consistent that for every cardinal $λ$ and every ordinal $α< λ$, there exists a centreless group $G$ such that (a) $τ(G) = α$; and (b) if $β$ is any ordinal such that $1 \leq β< λ$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $τ(G) = β$ in the corresponding generic extension $V^{P}$.
An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every natural number n greater than 1 and for every ordinal mu greater than or equal to aleph_n, we have (aleph_n, mu)-ITP.
Let kappa be the least ordinal alpha such that L_{alpha}(R) is admissible. Let A be the set of reals x such that x is ordinal definable in L_α(R), for some alpha<kappa. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the following theory: ZFC - Replacement + "There exists $ω$ Woodin cardinals which are cofinal in the ordinals." T has consistency strength weaker than that of the theory ZFC + "There exists omega Woodin cardinals", but stronger than that of the theory ZFC + "There exists n Woodin Cardinals", for each n. Let M be the canonic…
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality…
We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of "ordinal anticipating Laver functions" for strong unfoldability. With the help of these functions, we show that the principle $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ with the property that there exists a subset $z$ of $κ$ such that every sub…
We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the Kim-Pillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Ha…
Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $κ,λ$ are infinite cardinals such that $κ^{+++} \leq λ$, $κ^{<κ}=κ$ and $2^κ= κ^+$, and $η$ is an ordinal with $κ^+\leq η<κ^{++}$ and $cf(η) = κ^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = η+ 1$, - the cardinality of the $α$th level of $B$ is $κ$ for every $α<η$, - and the cardinality of the $η$th level of $B$ is $λ$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardi…
For any regular cardinal $κ$ and ordinal $η<κ^{++}$ it is consistent that $2^κ$ is as large as you wish, and every function $f:η\to [κ,2^κ]\cap Card$ with $f(α)=κ$ for $cf(α)<κ$ is the cardinal sequence of some locally compact scattered space.
We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbolΠ^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} | κ_\infty, Λ),$ where $\mathcal{M}_\infty$ is a direct limit of iterates of $M_{n+1}$, $δ_\infty$ is the least Woodin cardinal in $\mathcal{M}_\infty$…
The existence of End Elementary Extensions of models M of ZFC is related to the ordinal height of M, according to classical results due to Keisler, Morley and Silver. In this paper, we further investigate the connection between the height of M and the existence of End Elementary Extensions of M. In particular, we prove that the theory `ZFC + GCH + there exist measurable cardinals + all inaccessible non weakly compact cardinals are possible heights of models with no End Elementary Extensions' is consistent relative to the theory `ZFC + GCH + there exist measurable cardinals + the weakly compact…
We investigate the extent to which ultrapowers by normal measures on $κ$ can be correct about powersets $\mathcal{P}(λ)$ for $λ>κ$. We consider two versions of this questions, the capturing property $\mathrm{CP}(κ,λ)$ and the local capturing property $\mathrm{LCP}(κ,λ)$. $\mathrm{CP}(κ,λ)$ holds if there is an ultrapower by a normal measure on $κ$ which correctly computes $\mathcal{P}(λ)$. $\mathrm{LCP}(κ,λ)$ is a weakening of $\mathrm{CP}(κ,λ)$ which holds if every subset of $λ$ is contained in some ultrapower by a normal measure on $κ$. After examining the basic properties of these two notio…
We prove from the existence of a Mahlo cardinal the consistency of the statement that $2^ω= ω_3$ holds and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects to an ordinal less than $ω_2$ with cofinality $ω_1$.
Let GCH hold and let $j:V\longrightarrow M$ be a definable elementary embedding such that $crit(j)=κ$, $^κM\subseteq M$ and $κ^{++}=κ_{M}^{++}$. H. Woodin proved that there is a cofinality preserving generic extension in which $κ$ is measurable and GCH fails at it. This is done by using an Easton support iteration of Cohen forcings for blowing the power of every inaccessible $α\leqκ$ to $α^{++}$, and then adding another forcing on top of that. We show that it is enough to use the iterated forcing, and that the latter forcing is not needed. We will show this not only for the case where $κ^{++}=…
We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD. Nevertheless, the procedure resembles the one developed by Steel. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stationary class of lambda such that lambda is a limit of Woodin cardinals and the derived model at lambda satisfies AD the first member of the Solovay sequence is is less than Theta then there is a transitive model …
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists. Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an …
We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $ω_1$ and height $ω_2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $ω_1$ and height $ω_3$. Also, we prove a stepping up theorem that, for every cardinal $λ\geq ω_2$, permits us to construct from an LLSP space of width $ω_1$ and height $λ$ satisfying certain additional properties an LLSP space of width $ω_1$ and height $α$ for every ordinal $α< λ^+$. Then, we obt…
