# Introduction Among the most fundamental results in set-theory are Gödel's proofs of the consistency of the axiom of choice (AC) and the continuum hypothesis (CH) with the axioms of Zermelo-Fraenkel set-theory (ZF), using the universe of constructible sets - $L$. $L$ is defined as a cumulative hierarchy, beginning with the empty set, taking unions at limit stages, and successor stages consist of all the sets definable, using first-order logic, over the previous stage. A natural generalization of this construction is to strengthen the notion of definability -- from first-order logic to some extended logic. The result is a plethora of inner-models of ZF, which may share some of $Ls properties, while allowing objects and properties which are excluded from $L$, such as certain large cardinals. Models of this kind have been studied by Chang [[Chang - Sets constructible using Lκκ]] and Myhill-Scott [[Myhill, Scott - Ordinal definability]], and more recently a general framework was set-up by Kennedy, Magidor and Väänänen (abbreviated KMV) [[Kennedy, Magidor, Väänänen - Inner models from extended logics_ Part 1]], on which we rely here. # Definition ## Original definition 1. If $\mathcal{L}$ is a logic (always extending first-order logic), $M$ a set, then $\operatorname{Def}_{\mathcal{L}}\left(M\right)=\left\{ \left\{ a\in M\mid\left(M,\in\right)\vDash_{\mathcal{L}}\varphi\left(a,\bar{b}\right)\right\} \mid\varphi\in\mathcal{L};\,\bar{b}\in M^{<\omega}\right\}$ 2. $C\left(\mathcal{L}\right)$, the class of *$\mathcal{L}$-constructible sets*, is defined by induction: $\begin{aligned} L'_{0} & =\varnothing\\ L'_{\alpha+1} & =\operatorname{Def}_{\mathcal{L}}\left(L'_{\alpha}\right)\\ L_{\beta}' & =\bigcup_{\alpha<\beta}L_{\alpha}'\,\,\,\text{for limit }\beta \\ C\left(\mathcal{L}\right) & =\bigcup_{\alpha\in Ord}L'_{\alpha}\end{aligned}$ Note that if $\mathcal{L}$ is first-order logic, then we get exactly Gödel's [[Constructible universe]] $L$. **Theorem.** For any logic $\mathcal{L}$, $C\left(\mathcal{L}\right)$ is a transitive model of $\mathrm{ZF}$ containing all the ordinals. ## Revised definition It has been noted by Gabriel Goldberg that in the case of [[Inner models from extended logics#Stationary logic|stationary logic]], it is not clear whether the model defined as above satisfies the [[Choice principles|Axiom of Choice]]. To address this, KMV have revised the definition of $C(\mathcal{L})$ in [[Kennedy, Magidor, Väänänen - Inner models from extended logics_ Part 2]]. Suppose $\mathcal{L}^*$ is a logic the sentences of which are (coded by) natural numbers. We define the hierarchy $\left(J_\alpha^{\prime}\right), \alpha \in \operatorname{Lim}$, of sets constructible using $\mathcal{L}^*$ and the class $\operatorname{Tr}$, by transfinite double induction, as follows: $ \operatorname{Tr}=\left\{(\alpha, \varphi(\vec{a})):\left(J_\alpha^{\prime}, \in, \operatorname{Tr}\restriction\alpha) \models \varphi(\vec{a}), \varphi(\vec{x}) \in \mathcal{L}^*, \vec{a} \in J_\alpha^{\prime}, \alpha \in \operatorname{Lim}\right\}\right. $ where $ \operatorname{Tr}\restriction\alpha=\{(\beta, \psi(\vec{a})) \in \operatorname{Tr}: \beta \in \alpha \cap \operatorname{Lim}\} $ and $ \begin{cases}J_0^{\prime} & =\emptyset \\ J_{\alpha+\omega}^{\prime} & =\operatorname{rud}_{\operatorname{Tr}}\left(J_\alpha^{\prime} \cup\left\{J_\alpha^{\prime}\right\}\right) \\ J_{\omega \nu}^{\prime} & =\bigcup_{\alpha<\nu} J_{\omega \alpha}^{\prime}, \text { for } \nu \in \operatorname{Lim}\end{cases} $ Where the rudimentary closure operation $\operatorname{rud}_{\mathrm{Tr}}$ includes the operation $x \mapsto x \cap \operatorname{Tr}$. $C\left(\mathcal{L}^*\right)$ denotes the class $\bigcup_{\alpha \in \operatorname{Lim}} J_\alpha^{\prime}$ and $\mathbf{J}_\alpha^{\prime}$ denote the structure $\left(J_\alpha^{\prime}, \in, \operatorname{Tr} \upharpoonright \alpha\right)$. # Iterating the $C(\mathcal{L})$ construction As we mentioned above, it is possible that for some logic $\mathcal{L}$, $\left.C(\mathcal{L})\right.^{C(\mathcal{L})}\ne C(\mathcal{L})$ . This raises two questions for any extended logic -- first, under what circumstances can we get $\left.C(\mathcal{L})\right.^{C(\mathcal{L})}=C(\mathcal{L})$? And second, when this is not the case, how much can we iterate the construction? Formally >[!info] Definition >The *iterated $C(\mathcal{L})$ sequence* is defined by $\begin{aligned} \left.C(\mathcal{L})\right.^{0} & =V\\ \left.C(\mathcal{L})\right.^{\left(\alpha+1\right)} & =\left.C(\mathcal{L})\right.^{\left.C(\mathcal{L})\right.^{\alpha}}\,\text{ for any }\alpha \\ \left.C(\mathcal{L})\right.^{\alpha} & =\bigcap_{\beta<\alpha}\left.C(\mathcal{L})\right.^{\beta}\,\,\text{ for limit }\alpha \end{aligned}$ This type of construction was first investigated by McAloon [@mcaloon2] regarding $\mathrm{HOD}=C(\mathcal{L}^{2})$, where he showed: **Theorem.** It is equiconsistent with $\mathrm{ZFC}$ that there is a strictly decreasing sequence of iterated $\mathrm{HOD}$ of length $\omega$, and the intersection of the sequence can be either a model of $\mathrm{ZFC}$ or of $\mathrm{ZF+\neg AC}$. Somewhat later it was shown by Harrington (in unpublished notes, cf. [@ZADROZNY]) that the intersection might not even be a model of $\mathrm{ZF}$. Jech [@jech1975descending] used forcing with Suslin trees to show that it is possible to have a strictly decreasing sequence of iterated $\mathrm{HOD}$ of any arbitrary ordinal length. Zadrożny [@zadrozny1981transfinite] improved this to an ${\bf Ord}$ length sequence. So in the case of $\mathrm{HOD}$, "everything is possible". In [@ZADROZNY] Zadrożny generalized McAloon's method and gave a more flexible framework for coding sets by forcing, which he used to give another proof of this result. # Some known results [[Kennedy, Magidor, Väänänen - Inner models from extended logics - Part 1]] ## Infinitary logics >[!info] Definition >For regular cardinals $\lambda\leq\kappa$, $\mathcal{L}_{\kappa\lambda}$ is the logic which includes conjunctions and disjunctions of size lt;\kappa$ ($\bigwedge_{\xi<\gamma}$,$\bigvee_{\xi<\gamma}$ for $\gamma<\kappa$) and universal and existential quantification of size lt;\lambda$ ($\forall_{\xi<\gamma}$,$\exists_{\xi<\gamma}$ for $\gamma<\lambda$). > Replacing $\kappa$ or $\lambda$ by $\infty$ indicates "any size". > $\mathcal{L}_{\kappa\lambda}^{\omega}$ denotes the class of $\mathcal{L}_{\kappa\lambda}$ formulas with only finitely many free variables. The models $C\left(\mathcal{L}_{\kappa\kappa}\right)=C^{\kappa}$ constructed from the infinitary logics $\mathcal{L}_{\kappa\kappa}$ for $\kappa$ a regular cardinal were studied by Chang in [@changLkk], and the case of $\kappa=\omega_{1}$ is called "[[Chang's model]]". The model $C^{\kappa}$ can be characterized as the smallest transitive model of $\mathrm{ZF}$ closed under sequences of size lt;\kappa$. It has some similarity to $L$ by satisfying the axiom $V=C^{\kappa}$, and having some Condensation phenomena which allows one to prove that that if the Axiom of Choice holds in $C^{\kappa}$ then GCH holds on a cofinal class of cardinals, specifically that $2^{\nu^{<\kappa}}=\left(\nu^{<\kappa}\right)^{+}$ for any cardinal $\nu$. However, $C^{\kappa}$ might *not* satisfy the Axiom of Choice - Kunen showed in [@kunen-negAC] that if there are $\lambda^{+}$ many measurable cardinals (for $\lambda\geq\omega$ regular) then $C^{\lambda^{+}}\vDash\neg\mathrm{AC}$. Regarding large cardinals, essentially the same argument as in $L$ shows that $\kappa$ cannot be measurable in $C^{\kappa^{+}}$. However, if $V=L^{U}$ where $U$ is a normal ultrafilter on $\kappa$, we have $U\in C^{\kappa^{++}}$, so $C^{\kappa^{++}}=L^{U}$ and $\kappa$ is measurable there, so this bound is strict. >[!note] Remark >$C\left(\mathcal{L}_{\omega_{1}\omega}^{\omega}\right)=L\left(\mathbb{R}\right)$ >$C\left(\mathcal{L}_{\infty\omega}^{\omega}\right)=V$ ## Higher order logics $\mathcal{L}^{n}$, and their fragments, e.g. the classes of $\Sigma_{m}^{n}$, $\Pi_{m}^{n}$ and $\Delta_{m}^{n}$ formulas. Myhill and Scott [[Myhill, Scott - Ordinal definability]] have shown that for $n\geq2$, $C\left(\mathcal{L}^{n}\right)=\mathrm{HOD}$ - the inner model of [[HOD|hereditarily ordinal definable sets]], introduced by Gödel as well. $\mathrm{HOD}$ also satisfies ZFC, but in contrast to $L$, it is far less well-behaved. First, it might not satisfy $V=\mathrm{HOD}$ - McAloon ([[McAloon - Some applications of Cohen's method]], [[McAloon - On the sequence of models HODₙ]]) showed that in fact one can use forcing to construct a model in which there is a strictly descending sequence $\mathrm{HOD}\supsetneq\mathrm{HOD}^{\mathrm{HOD}}\supseteq\mathrm{HOD}^{\mathrm{HOD}^{\mathrm{HOD}}}\supseteq\dots$ (see [[Inner models from extended logics#Iterating the $C( mathcal{L})$ construction |discussion below]]). Second, $\mathrm{HOD}$ is extremely non-canonical - for example, every set can be forced to be in $\mathrm{HOD}$, and even more - every model of $\mathrm{ZFC}$ can be forced to be the $\mathrm{HOD}$ of another model (see [[Roguski - Extensions of models for ZFC to models for ZF + V = HOD with applications]]), so $\mathrm{HOD}$ can satisfy any consistent extension of $\mathrm{ZFC}$. However, $V=\mathrm{HOD}$ is relatively consistent with very large cardinals (see e.g.[[McAloon - Consistency results about ordinal definability]] [[Menas - On strong compactness and supercompactness]], [[Hamkins - The Wholeness Axioms and V=HOD]]), which is desirable. $\mathrm{HOD}$ is obtained also when weakening $\mathcal{L}^{2}$ to logics in which the second-order quantifiers range only on sets which are small relative to the model they live in, as well as when using only $\Sigma_{2}^{1}$ formulas. However, if we restrict only to $\Sigma_{1}^{1}$, we may get something new: Set $\mathrm{HOD}_{1}=C\left(\Sigma_{1}^{1}\right)$. Then it is consistent with $\mathrm{ZFC}$ that $\mathrm{HOD}_{1}\ne\mathrm{HOD}$, while many of the models $C\left(\mathcal{L}^{*}\right)$ discussed below are contained in $\mathrm{HOD}_{1}$, so it is relatively robust. In particular if $0^{\sharp}$ [[Zero sharp|exists]] then it is in $\mathrm{HOD}_{1}$, and $\mathrm{HOD}_{1}$ can accommodate large cardinals such as [[supercompact]] (if it exists). ## Generalized quantifiers We add to first order logic an additional $n$-place quantifier $Q$, so we obtain formulas of the form $Qx_{1},\dots,x_{n}\varphi\left(x_{1},\dots,x_{n}\right)$. The general semantics of $Q$ is given by a defining class $\mathcal{K}_{Q}$, and setting $\begin{gathered} \mathcal{M}\vDash Qx_{1},\dots,x_{n}\varphi\left(x_{1},\dots,x_{n},\bar{b}\right)\iff\\ \left(M,\left\{ \left(a_{1},\dots,a_{n}\right)\in M^{n}\mid\mathcal{M}\vDash\varphi\left(a_{1},\dots,a_{n},\bar{b}\right)\right\} \right)\in\mathcal{K}_{Q}\end{gathered}$ but usually we will define explicitly when $\mathcal{M}\vDash Qx_{1},\dots,x_{n}\varphi\left(x_{1},\dots,x_{n},\bar{b}\right)$ and $\mathcal{K}_{Q}$ will be implicit. These logics are denoted $\mathcal{L}\left(Q\right)$ and the models constructed with them $C\left(Q\right)$. ### Cardinality quantifiers >[!info] Definition >For an ordinal $\alpha$, $\mathcal{M}\vDash Q_{\alpha}x\varphi\left(x,\bar{b}\right)\iff\left|\left\{ a\in M\mid\mathcal{M}\vDash\varphi\left(a,\bar{b}\right)\right\} \right|\geq\aleph_{\alpha}$ ### The Magidor-Malitz quantifier >[!info] Definition >For an ordinal $\alpha$ and integer $n$, $\begin{gathered} \mathcal{M}\vDash Q_{\alpha}^{\mathrm{MM},n}x_{1},\dots,x_{n}\varphi\left(x_{1},\dots,x_{n},\bar{b}\right)\iff\\ \exists X\subseteq M\left(\left|X\right|\geq\aleph_{\alpha}\land\forall a_{1},\dots,a_{n}\in X:\mathcal{M}\vDash\varphi\left(a_{1},\dots,a_{n},\bar{b}\right)\right) \end{gathered}$ It is possible to force over $V=L$ to obtain a model with $C\left(Q_{\omega_{1}}^{\mathrm{MM},2}\right)\ne L$. However, if $0^{\#}$ exists, then $C\left(Q_{\alpha}^{\mathrm{MM},<\omega}\right)=L$ for every $\alpha$. ### Cofinality quantifiers >[!info] Definition >For a regular $\kappa$, >$\begin{align} \mathcal{M}\vDash Q_{\kappa}^{\mathrm{cf}}xy\varphi\left(x,y,\bar{b}\right)\iff{\left\{ \left(c,d\right)\in M\mid\mathcal{M}\vDash\varphi\left(c,d,\bar{b}\right)\right\} } \\ \text{is a linear order of cofinality }\kappa \end{align}$ Denote $C\left(Q_{\lambda}^{\mathrm{cf}}\right)$ by $C_{\lambda}^{*}$ , $C^{*}:=C_{\omega}^{*}$. These models may be more robust than $L$: **Theorem.** - If $0^{\#}$ exists then $0^{\#}\in C_{\lambda}^{*}$ - The Dodd-Jensen [[Core models|core model]] is contained in $C^{*}$ - If there is an inner model for a [[Measurable|measurable cardinal]] then there is one contained in $C^{*}$. Scott's proof that measurable cardinals contradicts $V=L$ can be applied to obtain that a measurable $\kappa$ implies $V\ne C_{\lambda}^{*}$ for any $\lambda<\kappa$, but it is still unknown whether there can be a measurable cardinal in $C_{\lambda}^{*}$ when $V=C_{\lambda}^{*}$ fails. These models might not be idempotent - one may force over a model of $V=L$ to obtain a model in which $\left(C^{*}\right)^{C^{*}}\ne C^{*}$. However, if $\left(C^{*}\right)^{C^{*}}=C^{*}$ then $C^{*}\vDash2^{\aleph_{\alpha}}=\aleph_{\alpha}^{+}$ for every $\alpha\geq1$, and $2^{\aleph_{0}}\leq\aleph_{2}$. This is shown in a way similar to the proof of GCH in $L$, while essentially using the fact that $\mathcal{L}\left(Q_{\omega}^{\mathrm{cf}}\right)$ has the Löwenheim-Skolem-Tarski property (LST) down to $\omega_{1}$. $\mathrm{CH}$ itself cannot be obtained in general, as one can force $V=C^{*}+2^{\aleph_{0}}=\aleph_{2}$ over a model of $V=L$+"there is an inaccessible cardinal". Without the assumption $\left(C^{*}\right)^{C^{*}}=C^{*}$ things are more flexible - one may force over $V=L$ to obtain $C^{*}\vDash2^{\omega}=\kappa$ for any $\kappa$ of uncountable cofinality. A main result of [[Kennedy, Magidor, Väänänen - Inner models from extended logics_ Part 1]] is that under large cardinals, the theory of $C^{*}$ is invariant by set forcing, and independent of the cardinality used. More precisely: **Theorem.** If the is a proper class of Woodin cardinals, $\mathbb{P}$ is a forcing notion, $G\subseteq\mathbb{P}$ generic, $\lambda$ regular, then $Th\left(\left(C^{*}\right)^{V}\right)=Th\left(\left(C^{*}\right)^{V\left[G\right]}\right)=Th\left(\left(C_{<\lambda}^{*}\right)^{V}\right)$ So it is canonical in some sense. However not much is known of this theory. For example, it is not known if $\mathrm{CH}$ hold in $C^{*}$ under the above assumption or not. Although the theory is the same, these models can very well be different, and may change under forcing. ### The Härtig quantifier [[Härtig quantifier]] The model $C\left(I\right)$ shows similar robustness -- KMV show that it also contains $0^{\#}$ if it exists, the Dodd-Jensen core model, and an inner model for a measurable, if there is such inner model. They also show that if $V=L^{\mu}$ then $V=C\left(I\right)$, while the other hand, if there is no inner model for a measurable cardinal, then $C\left(I\right)$ has a covering theorem -- every uncountable set of ordinals is contained in a set from $C\left(I\right)$ of the same cardinality. Recently Welch [[Welch - C_ in L[E]-Models below Oᵏ]] extended this analysis: let $O^{k}$ (O-kukri) be the statement that there is an elementary embedding of an inner model with a proper class of measurables to itself, then: - If $V=L[E]$, where $E$ is a coherent sequence of extenders then $\neg O^{k}\iff V=C(I)$ - $\neg O^{k}\iff$ $C(I)$ satisfies weak covering -- for every singular cardinal $\tau$, $\tau^{+}=\left(\tau^{+}\right)^{C(I)}$. - If $O^{k}$ exists, then it is *not* an element of $C(I)$. - If $O^{k}$ exists then $C(I)=L[E][c]$ where $E$ is a sequence of (class many) measures, non of which is a limit of measurables, and $c$ is generic for a class Prikry forcing corresponding to $E$ -- for any $\kappa$ on which $E(\kappa)$ is a measure, $c(\kappa)$ is a Prikry sequence. An interesting feature of $\mathcal{L}\left(I\right)$ is that the LST-property, which allows to prove results similar to those in $L$, has large cardinal strength. ## Stationary logic [[Kennedy, Magidor, Väänänen - Inner models from extended logics Part 2]] ![[Stationary logic#^def-logic]] The model $C\left(\mathtt{aa}\right)$ obtained from stationary logic contains $C^{*}$, since notion of "being of cofinality $\omegaquot; is $\mathcal{L}\left(\mathtt{aa}\right)$-definable. In is a model of ZF, but recent results show it might not always be a model of AC. KMV dedicate a second part of their paper [@IMEL2] to the study of this model and its variants. Under large cardinal assumptions it is possible to show that this model is much more robust: assuming one of the following - (a) there is a proper class Woodin cardinals; (b) there is a supercompact cardinal; (c) $\mathrm{PFA}^{++}$; then every uncountable regular cardinal is measurable in $C\left(\mathtt{aa}\right)$. Another result is that a proper class of measurable Woodin cardinals implies that the theory of $C\left(\mathtt{aa}\right)$ is invariant under set-forcing. Regarding GCH, KMV have recently proven that for every $\alpha<\omega_{1}^{V}$, $C\left(\mathtt{aa}\right)\vDash2^{\aleph_{\alpha}}=\aleph_{\alpha}^{+}$, and from recent unpublisher results of Goldberg and Steel it seems that in fact GCH holds everywhere. Still much of the study scheme remains open. Note that the forcing invariance implies that at least some of these questions are settled by large cardinal assumptions.