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Łoś's theorem is a fundamental theorem in model theory (a branch of mathematical logic).

Historical question: What was Łoś's original motivation to define ultraproducts and prove Łoś's theorem? Which kind of problems let to its discovery?

Modern applications include an elegant proof of the compactness theorem and the construction of saturated models. However, often modern applications do not coincide with the original motivation that led to its discovery.

Apparently this is Łoś's original paper. In their model theory book, Chang and Keisler write:

The ultraproduct construction goes back to the work of Skolem (1934).

So it seems this is an important precursor to Łoś's construction. Though Łoś doesn't cite it, but maybe he cites papers that cite it, I don't know.

Reformulation of the question: If somebody here is able to read French and German, or already knows about the history of ultraproducts, can he summarize the context in which Łoś's and Skolem's paper were written?

The language of Skolem's paper is so far removed from modern logic that I can't make head nor tail of it. His title translates to "On the non-characterizability of number sequence via finite or countably infinite many statements only containing number variables". He talks a lot about "arithmetic functions" -- what has this to do with model theory?

It's quite fascinating to learn about the history of some fundamental mathematical idea.

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