Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}\_{\omega ,\omega}$). One can produce such a universe in infinitary logics in the same way using corresponding notions of formulas and definability. Obviously $L$ becomes larger when the logic has more expression power.


For each cardinal $\kappa$ define $L\_{\kappa}$ to be Gödel's constructible universe in the infinitary logic $\mathcal{L}\_{\kappa,\kappa}$ and $L\_{\infty}$ is Gödel's constructible universe in $\mathcal{L}\_{\infty,\infty}$.


(1) Is $L\_{\kappa}$ a model of $ZFC$ for each cardinal $\kappa$? What about $ZFC+GCH$?


(2) What is $L\_{\infty}$?


(3) Is there a (possibly large) cardinal $\kappa$ such that $L\_{\kappa}$ is Dodd-Jensen core model, $L[U]$, $HOD$, etc?


(4) What are the consistency strengths of the existence of non-trivial elementary embeddings from $\langle L\_{\kappa},\in\rangle$ to itself for different $\kappa$s in the sense of infinitary logic $\mathcal{L}\_{\kappa,\kappa}$? 


Note that by Prof. Hamkins' answer for $L\_{\infty}$ finally it reach Kunen's inconsistency but what about a given cardinal $\kappa$? Are all these consistency strengths for different cardinals bounded by some large cardinal axiom and there is a gape between consistency strength of the existence of a non-trivial elementary embedding from $\langle L\_{\infty},\in\rangle$ to itself and consistency strengths of the existence of such elementary embeddings for $L\_{\kappa}$s?


(5) If there is a cardinal $\kappa$ such that $L\_{\kappa}=HOD$, is it possible to determine consistency strength of the existence of a non-trivial (first-order) elementary embedding from $\langle HOD,\in\rangle$ to itself by analyzing the growth speed of the consistency strength of existence of such embeddings for $\langle L\_{\lambda}, \in\rangle$s in $\lambda <\kappa$?


(6) What is $L\_{\kappa}$ for the least strongly compact cardinal $\kappa$?



Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)
**Theorem.** $L\_\infty$ is the entire set-theoretic universe $V$.


Proof. I claim that every set will arise in the construction process, because eventually it will become explicitly definable by a formula. In infinitary logic, there are far more than only countably many formulas, and one can cook up a formula to define a specific set, by using the formulas that define its elements. What I claim specifically is that for every set $a$ there is a ${\cal L}\_{\infty,\infty}$ formula $\psi\_a(x)$, such that in any transitive set $M$ with $a\subset M$ we have $a=\{ x\mid M\models\psi\_a(x)\}$. Suppose that this is true for each $a\in A$. Consider the formula $$\psi\_A(u)=\bigvee\_{a\in A}(u=\{x\mid \psi\_a(x)\}).$$
In any transitive $M$ with $A\subset M$, it follows that $\psi\_A(u)$ will hold if and only if $u=a$ for some $a\in A$. Thus, $A=\{u\mid M\models \psi\_A(u)\}$, and so $A$ is also definable. Thus, by $\in$-induction, we've verified that every set is definable, and so every set in $V$ eventually arises in your universe $L\_\infty$. QED


This argument is similar to the fact that if one undertakes the constructible universe using second-order logic, rather than infinitary logic, the result is $\text{HOD}$.


**Theorem.**(Myhill & Scott) The constructible universe in second-order logic is the same as HOD. $$L\_{SO}=\text{HOD}.$$


Proof. Any set that appears in $L\_{SO}$ is ordinal definable, and so $L\_{SO}\subset \text{HOD}$. Conversely, if $A$ is a set of ordinals in $\text{HOD}$, then it is ordinal definable in some $V\_\alpha$, and so once $L\_{SO}$ has constructed up to some stage $\theta$ of size at least $|V\_\alpha|$, then in second order logic we can define $A$ as a subset by saying "there is a relation on $\theta$ which makes it isomorphic to $\langle V\_\alpha,\in\rangle$, such that the formula is true for the corresponding ordinals. That is, the second-order logic allows us to summon a copy of $V\_\alpha$ and run the definitions inside it. QED


For a reference, see [Myhill, Scott, Ordinal definability. 1971 Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., (1967) pp. 271–278 Amer. Math. Soc., Providence, R.I.](http://www.ams.org/mathscinet-getitem?mr=281603) .


Your models $L\_\kappa$ arise in Chang's paper in the very same journal issue:



> 
> C.C. Chang, [Sets constructible using $L\_{\kappa,\kappa}$, 1971 Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) pp. 1–8 Amer. Math. Soc., Providence, R.I.](http://www.ams.org/mathscinet-getitem?mr=280357)
> 
> 
> **MR Abstract.** Let $C\_\alpha^\omega$ be the αth level in the Gödel ramified hierarchy of constructible sets. The author studies the hierarchy $C\_α^κ$ that results on using $L\_{κκ}$ definability (in place of $L\_{ωω}$ definability) in generating this hierarchy, where κ is a regular cardinal. The class $C\_κ=⋃\_αC\_α^κ$ is a model of ZF; in fact, it is the smallest transitive model containing all the ordinals closed under <κ-termed sequences (Theorems I and II). It need not be a model of the axiom of choice (Theorem IV). Other results show what happens to the GCH and Scott's result on measurable cardinals in this model. These results depend on a generalization of Gödel's collapsing lemma (Theorem V). Finally, the author shows how to improve results about indiscernibles in $C\_ω$ by using infinitary formulas. In particular, he shows that if there is a Ramsey cardinal then $C\_{ω\_1ω}$ is an $L\_{ω\_1ω}$-elementary substructure of $C\_ω$, thus improving Silver's result.
> 
> 
> 


The case $\kappa=\omega\_1$ is known as the *Chang* model.


