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) "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, and etc.

But when Godel proved that a first-order sentence has a finite proof if and only if it holds in every "model" -- what sort of model 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 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:

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