48 pages. Added hypotheses to Thms 1.4 and 1.6, to bridge minor gaps in their former "proofs". Appeal to [6, Thm 0.2] (in paper's references) for Thm 4.2 clarified. Last part of proof of Thm 4.7 clarified. Added Lem 8.14 and strengthened Lem 8.13, to establish what was claimed to be an "immediate consequence" of STH in previous version. Other minor corrections and improvements in exposition
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.
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 fi…
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_…
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…
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\rfl…
We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.
We consider functions of natural numbers which allow a combinatorial interpretation as density functions (speed) of classes of relational structures, s uch as Fibonacci numbers, Bell numbers, Catalan numbers and the like. Many of these functions satisfy a linear recurrence relation over $\mathbb Z$ or ${\mathbb Z}_m$ and allow an interpretation as counting the number of relations satisfying a property expressible in Monadic Second Order Logic (MSOL). C. Blatter and E. Specker (1981) showed that if such a function $f$ counts the number of binary relations satisfying a property expressible in MS…
We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}_{alg}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in $\mathbb{R}_{an,exp}$ with respect to this structure. As a consequence we reprove that the zero loci of admissible normal functions are algebraic.
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.
In the course of many mathematical developments involving 'number systems' like $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb {C}$ etc., it sometimes becomes necessary to abstract away and study certain properties of the number system in question so that we may better understand objects having these properties in a more general setting -- a relevant example for this paper would be Hausdorffs $η_\varepsilon$-fields, objects abstracted from the property that there are no two disjoint intervals in $\mathbb{R}$ of cardinality $α\leq \aleph_0$ whose union is $\mathbb{R}$. My goal will be…
There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\ma…
It is an old problem in the area of Diophantine definability to determine whether $\mathbb{Q}$ is Diophantine in $\mathbb{Q}(z)$. We provide a positive answer conditional on two standard conjectures on elliptic surfaces.
We study the model theory of the $2$-sorted structure $(\mathbb{F}, \mathbb{C};χ)$, where $\mathbb{F}$ is an algebraic closure of a finite field of characteristic $p$, $\mathbb{C}$ is the field of complex numbers and $χ: \mathbb{F} \to \mathbb{C}$ is an injective, multiplication preserving map. We obtain an axiomatization $\mathrm{ACFC}_p$ of $\mathrm{Th}(\mathbb{F},\mathbb{C};χ)$ in a suitable language $L$, classify the models of $\mathrm{ACFC}_p$ up to isomorphism, prove a modified model companion result, give various descriptions of definable sets inside a model of $\mathrm{ACFC}_p$, and de…
We present a class forcing notion $\mathbb M(η)$, uniformly definable for ordinals $η$, which forces the ground model to be the $η$-th inner mantle of the extension, in which the sequence of inner mantles has length at least $η$. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that $\mathbb M(η)$ forces the ground model to be the $η$-th iterated HOD of the extension, where the sequence of iterated HODs has length at least $η$. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any tw…
In arXiv:2208.12944 it is shown that an ordinal $\sup_{N<ω}ψ_{Ω_{1}}(\varepsilon_{Ω_{\mathbb{S}+N}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory ${\sf KP}\ell^{r}+(M\prec_{Σ_{1}}V)$. In this paper we show that a second order arithmetic $Σ^{1-}_{2}\mbox{-CA}+Π^{1}_{1}\mbox{-CA}_{0}$ proves the wellfoundedness up to $ψ_{Ω_{1}}(\varepsilon_{Ω_{\mathbb{S}+N+1}})$ for each $N$. It is easy to interpret $Σ^{1-}_{2}\mbox{-CA}+Π^{1}_{1}\mbox{-CA}_{0}$ in ${\sf KP}\ell^{r}+(M\prec_{Σ_{1}}V)$.
The assertion that every definable set has a definable element is equivalent over ZF to the principle $V=\text{HOD}$, and indeed, we prove, so is the assertion merely that every $Π_2$-definable set has an ordinal-definable element. Meanwhile, every model of ZFC has a forcing extension satisfying $V\neq\text{HOD}$ in which every $Σ_2$-definable set has an ordinal-definable element. Similar results hold for $\text{HOD}(\mathbb{R})$ and $\text{HOD}(\text{Ord}^ω)$ and other natural instances of $\text{HOD}(X)$.
Under a mild definability assumption, we characterize the family of Borel actions $Γ\curvearrowright X$ of tsi Polish groups on Polish spaces that can be decomposed into countably-many actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of $1_Γ$. In the special case that $Γ$ is non-archimedean, it follows that there is such a decomposition if and only if there is no continuous embedding of $\mathbb{E}_0^{\mathbb{N}}$ into $E_Γ^X$.
This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.
In previous work, the author has shown that $Π^1_1$-induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of J.-Y. Girard's dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called $2$-ptykes. We show that $Π^1_2$-induction along $\mathbb N$ is equivalent to the existence of fixed points for…
Brouwer (1927) claimed that every function from the Baire space to natural numbers is induced by a neighbourhood function whose domain admits bar induction. We show that Brouwer's claim is provable in Heyting arithmetic in all finite types ($\mathrm{HA}^ω$) for definable functions of the system. The proof does not rely on elaborate proof theoretic methods such as normalisation or ordinal analysis. Instead, we internalise in $\mathrm{HA}^ω$ the dialogue tree interpretation of Gödel's system T due to Escardó (2013). The interpretation determines a syntactic translation of terms, which yields a n…
