TITLE: Completeness for Infinitary Logic?
QUESTION [7 upvotes]: I have heard a rumor that there is a proof system for certain infinitary logics, given by Carol Karp (?) in her thesis, but I can't find a copy.  The result that I'm told exists is the following:

A sentence $\phi$, from $L_{\omega_1\omega}$, has a model if and only if it is consistent; that is, there is no proof (in the above sense) from $\phi$ of a false sentence (e.g. $\exists x (x\not=x)$)
This is not true for general $L_{\infty\omega}$ sentences; that is, there is a consistent sentence with no model. (The converse should be trivial; any sentence with a model should be consistent)

What I would really appreciate is a description of this proof system.  I thought I could "guess" it on my own, but I guess I can't.  I suspect, once given, that the proofs of the above theorems are relatively easy.  In fact I have the sentence which is consistent with no models, although clearly I don't have a precise argument for consistency without the proof system.
I'm aware of some model existence theorems (e.g. Makkai's) which are probably equivalent to the above.  But I'm really hunting for the proof system.

REPLY [3 votes]: I see this is a pretty old question, but I recently checked out a copy of Karp's book from the library. 
These are from Chapters 5 & 11, if you manage to find a copy. The axiom schemes of the (Hilbert-style) system for $L_{\alpha \beta}$ ($\alpha$ is regular, infinite; $\beta$ is $0$ or $\omega \leq \beta \leq \alpha$) are as follows: 
(A1) $A \to (B \to A)$
(A2) $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$
(A3) $(\neg{A} \to \neg{B}) \to (B \to A)$
(A4. 1, $\delta$) $$\left({\Large\land}_{\xi < \delta}\, (A_\delta \to A_{\xi})\right) \to \left(A_\delta \to {\Large\land}_{\xi < \delta} \, A_{\xi}\right)$$ where $\{A_\xi: \xi < \delta\}$ is a sequence and $0 < \delta < \alpha$. 
(A4. 2, $\delta$, $\nu$) $$\left({\Large\land}_{\xi < \delta}\, A_{\xi}\right) \to A_\nu$$
where $\{A_\xi: \xi < \delta\}$ is a sequence, $0 < \delta < \alpha$, and $\nu < \delta$.
(I should mention that if we stop here and take modus ponens and conjunction (see below) as rules of inference, we get a proof system for the propositional language  $L_\alpha$.)
(A5) For $\overline{x} = \langle{x_{\xi}: \xi < \gamma\rangle}$ a sequence of variables with $\gamma < \beta$ and $\forall \overline{x}$ the abbreviation for $\forall x_{0} \dots \forall x_{\xi} \dots$: 
$$\forall \overline{x} (A \to B) \to (A \to \forall \overline{x} B)$$
provided no variable in $\overline{x}$ is free in $A$.
(A6) Abbreviations are as in (A5); $f$ is any function from $\{x_\xi: \xi < \gamma\}$ to $L$-terms and $\mathsf{SF}_{f}^{\overline{x}}(A)$ is the result of replacing each $x_\xi$ in $A$ with $f(x_\xi)$: 
$$\forall \overline{x} A \to \mathsf{SF}_{f}^{\overline{x}}(A)$$ provided that $A$ has no free occurrences of any $x_\xi$ bound by $\mathsf{FV}(f(x))$.
And as inference rules: 
(MP) From $A$, $A \to B$ infer $B$.
(Conj) From $A_\xi$ for all $\xi < \delta$ for some $0 < \delta < \alpha$, infer ${\Large \land}_{\xi < \delta} A_\xi$.
(Gen)  Where $\overline{y}$ and $\overline{x}$ are $\gamma$-sequences for $\gamma < \beta$: From $A$, infer $\forall \overline{x} \mathsf{SF}^{\overline{y}}(A)$ provided that $\overline{y} = \overline{x}$ or the variables of $\overline{y}$ are not among $\mathsf{FV}(A)$. 

And then Karp gives other axiom schemes which I think are only needed for predicate logics with infinitary atomic formulas. I'd have to finish reading the text to be sure.
Edit: I forgot the equality axioms. Here's a paper with Karp's axiomatization in full.