TITLE: Set theory bootstrapping
QUESTION [11 upvotes]: Let $\mathcal{L}$ be the first order language of ZFC set theory, and let $\mathcal{L}_{\infty,\infty}$ be the usual infinitary extension of the language allowing arbitrary long disjunctions/conjunctions and quantifications.
What happens if one replaces the usual axiom schema of replacement for ZFC by a new schema over (the class of) formulas from $\mathcal{L}_{\infty,\infty}$ instead of from the first-order theory?  Does one get a "standard" model of set theory?
Do set theorists believe that this improved version of replacement is true?  If so, why still work over the less expressive language $\mathcal{L}$?  If not, why not?
Does the connection between replacement and transfinite induction completely disappear under this extension?
Motivation: I was thinking about Skolem's paradox, and thought that working in $\mathcal{L}_{\infty,\infty}$ might resolve part of the problem---namely that first order language just wasn't expressive enough to talk about important aspects of big sets.  This seemed to be backed up by other things I read.  But then, it struck me that if we are working with a more expressive language, why limit our axioms to the simpler language?

REPLY [6 votes]: The idea of studying ZF and its subsystems formulated in infinitary languages, to my knowledge, seems to have begun and ended with the work of Klaus Gloede, in the 1970s. 
The following paper of Gloede provides a useful synopsis of his work, which began with his doctoral dissertation (it is available here behind a paywall, the first two pages are freely visible, they consist of the table of contents and the first page of the paper). 
K. Gloede, Set theory in infinitary languages. ⊨ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), pp. 311–362. Lecture Notes in Math., Vol. 499, Springer, Berlin, 1975.