TITLE: What "metatheory" did early set theory/logic researchers use to prove semantic results?
QUESTION [25 upvotes]: Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic.
The modern approach seems to be, usually, to interpret a "model" specifically as a set in some other (typically first-order) "set metatheory." So when we talk about a model of PA, for instance, we typically mean that we are formalizing it as a subtheory of something like first-order ZFC. Models of ZFC can be formalized in stronger set theories, such as those obtained by adding large cardinals, etc.
But when Godel proved that a first-order sentence has a finite proof if and only if it holds in every "model" -- what was he talking about? Likewise, how can we understand the Löwenheim-Skolem theorem if models didn't even exist at the time?
It is clear that these researchers were not talking about using first-order ZFC as a metatheory, as that theory didn't even gain popularity until after Cohen's work on forcing in the 60s. Likewise, NBG set theory had not yet been formalized. And yet, they were obviously talking about something. Did they have a different notion of semantics than the modern set-theoretic one?
In closing, two questions:

In general, how did early researchers (let's say pre-Cohen) formalize semantic concepts such as these?
Have any of these original works been translated into English, just to see directly how they treated semantics?

REPLY [9 votes]: To address the main question about Gödel's original proof of the completeness theorem, which as Timothy Chow explains can be found in Gödel's Collected Works, he is using the notion of “Erfüllungssystem”, which means something like "satisfactory system". It is a structure intended to satisfy a formula $\phi$ that is not refutable (i.e., such that $\neg \phi$ is not provable), whose underlying set is the natural numbers. He then extends his theorem for countable theories proving the compactness theorem.
The structure is built as what we would now call a filtered colimit of a countable chain of finite structures. This is done through the use of König's lemma, by first building a finitely branching tree of finite structures with the property that the there are structures at any finite level and the satisfaction in a given structure at level $n$ restricts to the satisfaction at the immediate predecessor. Then there is a cofinal branch through which he builds the model of $\phi$ on the structure given by the natural numbers.
It is worth mentioning that as an immediate consequence of his proof he gets the downward Löwenheim-Skolem theorem, since he builds a model on the structure whose underlying set is the natural numbers for any countable set of satisfiable formulas. Löwenheim's original "proof" had actually some flaws which later Skolem fixed.

REPLY [8 votes]: As Andreas Blass pointed out, the meta-theory can be ordinary mathematics, at least in theory. In practice, without an explicit meta-theory, authority figures decide what is allowed, and what not. Tarski (like Cantor before him) learned this lesson the hard way, as can be read in accounts of Tarski's theorem about choice from 1924:

... when he tried to publish the theorem in Comptes Rendus de l'Académie des Sciences Paris, Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest.

It is no surprise that the modern notion of model and meta-theory are due to Tarski (and his colleague Robert Vaught) from 1956. But Tarksi already presented a "non-modern" notion of meta-theory in 1933, see the SEP entry on Tarski's Truth Definitions.
The issue is a little bit complex, but here is my own summary of the difference between those two notions for a start:

If I understood it correctly, for the 1933 version, the model (i.e. the algebraic structure about which we talk) is part of the meta-language and not mentioned separately. The assignment of objects to variables on the other hand is what can satisfy a given formula. A formula is (defined to be) true if it is satisfied by all possible assignments of objects to variables.
The 1956 version is treated less explicitly in the linked SEP entry, but it is hinted at that the model is no longer an implicit part of the meta-language, but an explicit object from set-theory. A model can satisfy a given formula (or sentence), similar to how an "assignment of objects to variables" could satisfy a given formula for the 1933 version. But the text also hints that the 1956 now relies stronger on an underlying set-theory, while the 1933 explicitly tried to minimize "the set-theoretic requirements of the truth definition".