If a version of GCH holds for Chang's $\kappa$-constructibility, does a version of GCH hold for $L_{\infty}$? In C.C. Chang's paper "Sets Constructible Using $L_{\kappa \kappa}$" one can "deduce a version of the GCH, theorem VI [(iv)--my comment], assuming that all sets are $\kappa$-constructible."  Now $L_{\infty}$ is Goedel's constructible universe in $\mathcal{L}_{\infty \infty}$.  In his answer to the MathOverflow question "Goedel's Constructible Universe in Infinitary Logics", Prof Hamkins proves the following:
$L_{\infty}$=V.
Since this is tantamount to saying that 'all sets are $\infty$-constructible', can one deduce (as one can if all sets are assumed to be $\kappa$-constructible) that a version of GCH holds for $L_{\infty}$ (=V)?  If so, what version of GCH would this be?  If not, why does a version of GCH hold for the assumption that all sets are $\kappa$-constructible while for $L_{\infty}$=V the GCH is independent?
 A: Assuming that (choice holds and) $\mathrm{V}=C^\kappa$, we can prove the version of $\mathsf{GCH}$ stated in the paper, that for every $\nu>1$, $$2^{\nu^{<\kappa}}=(\nu^{<\kappa})^+,$$ since we can adapt the proof of $\mathsf{GCH}$ in $L$.
That argument depends on condensation, and the appropriate version of condensation is given in Theorem V of the paper, that $\kappa$-elementary substructures of initial segments of $C^\kappa$ that contain $\kappa$ as an element are themselves isomorphic to initial segments of $C^\kappa$.
This is used to bound the number of subsets of a cardinal $\lambda$ simply because any such subset should appear in an initial segment $C^\kappa_\eta$ of $C^\kappa$ where $\eta$ is not too large relative to $\lambda$ (as appropriate versions of the Löwenheim-Skolem theorem allow us to form "small" $\kappa$-elementary substructures). This gives us an upper bound on $2^\lambda$ in terms of the sizes of the initial stages $C^\kappa_\eta$. In turn, these sizes are bounded in terms of ($\eta$ and) the number possible formulas in the language. The latter number depends explicitly on $\kappa$.
It is then an easy matter to find cardinals $\lambda$ (that depend on $\kappa$) where this upper bound is realized, and this is what leads to the trace of $\mathsf{GCH}$ stated in the paper (in Theorem VI.(iv), as you indicate).
Note however that no version of this argument survives in $C^\infty=\mathrm{V}$, since there is no bound on the number of formulas in the language, so there is no a priori bound on the sizes of the initial segments $C^\infty_\eta$. In fact, it is consistent that $\mathsf{GCH}$ fails everywhere in $\mathrm{V}$.

Using pcf techniques, Shelah has found (what one could argue is) a version of $\mathsf{GCH}$ that actually holds provably in $\mathsf{ZFC}$, namely:

Theorem (Shelah). For $\lambda$ and $\kappa$ infinite cardinals, let $\lambda^{[\kappa]}$ denote the smallest possible size of a family $\mathcal F$ of $\kappa$-sized subsets of $\lambda$ such that any $\kappa$-sized subset of $\lambda$ is contained in the union of fewer than $\kappa$ many members of $\mathcal F$.
For any uncountable strong limit cardinal $\mu$, it is the case that for any $\lambda\ge\mu$ there is a $\theta<\mu$ such that for all $\kappa\in[\theta,\mu]$, we have  $$ \lambda^{[\kappa]}=\lambda. $$

Note however that this is far from the standard formulations of fragments of $\mathsf{GCH}$, and the known proofs of this result are delicate combinatorial arguments that do not relate to $\mathcal L_{\infty\infty}$ or appeal to either the fact that $\mathrm{V}=C^\infty$, or to adaptations of the condensation argument discussed above.
