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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 $\kappa$?

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You might find the recent work of Kennedy-Magidor-Vannanen interesting, by the way. They talk about generalized constructions using various quantifiers rather than infinitary logic. – Asaf Karagila Feb 7 '14 at 16:32
up vote 13 down vote accepted

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. .

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.

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.

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It is a really interesting observation that in this dual approach for building Godel's costructible universe by strengthening the expression power of our logic using longer formulas we will reach $V$ and if we increase the expression power of the logic by assuming wider range for quantifiers we will reach $HOD$. It seems $HOD$ and $V$ are some kind of limit $L$ in limit logics (The first one in $\mathcal{L}_{\infty,\infty}$ and the next one in $SO$). Maybe this means one can prove a dual form of Kunen's inconsistency theorem for $HOD$ too. – user46667 Feb 7 '14 at 17:15
Is it possible to find a proof of Kunen's inconsistency theorem in terms of infinitary logic by analyzing the structure of $L_{\infty}$ ($=V$)? If yes, is it possible to reformulate this proof for similar result about $L_{SO}$ ($=HOD$)? Maybe there is no non-trivial self-elementary embedding for $V$ because $V$ is $L$ of a too powerful logic like $\mathcal{L}_{\infty,\infty}$ as well as $HOD$ which is $L$ of another too powerful logic like $SO$. – user46667 Feb 7 '14 at 17:26
Joel, it seems your argument also shows that $L_\kappa$ (in Gina's sense) is $L(V_\kappa)$; is this correct? – Noah Schweber Feb 7 '14 at 17:46
@NoahS, since $V_\kappa$ could be much larger than $\kappa$ in size, I think $L([\kappa]^{<\kappa})$ is closer to being right. I'd have to think more about it. – Joel David Hamkins Feb 7 '14 at 17:53
Great! Chang's paper is a very nice reference that exactly matches my questions. Thank you very much. I don't know if Chang (or others) investigated around possible relations between these generalized constructible universes and Kunen inconsistency and HOD problem or not. – user46667 Feb 7 '14 at 18:53

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