Why does model theory treat well-ordered but uncountable languages?

Question

In model theory, results such as the Compactness theorem and the Löwenheim–Skolem theorem hold true, not only in countable languages, but in well-ordered languages.

I'm not questioning this idea in principle, but I am wondering why it is significant. That is, besides the fact that mathematical theorems ought to be as general as possible, what is the practical effect of extending such theorems to well-ordered languages? Are there even any examples of uncountable but well-ordered languages? If so, what do we gain by a study of such languages?

Answer 1

Now in the usual axioms for set theory every set is well-orderable - so "all well-orderable languages" just amounts to "all languages," and you're asking for examples of uncountable languages which are somehow useful. Nonstandard analysis provides one such example, where we look at a language naming every function on the real numbers (so this is a language of size $2^{2^{\aleph_0}}$, much larger than necessary). Another type of example is gotten by taking a structure $\mathcal{M}$, some subset $A$ of the domain of $\mathcal{M}$ (possibly the whole thing), and adding a constant symbol for each element of $A$; this type of expansion occurs frequently within model theory, even to study structures in "small" languages.

However, if we weaken our ambient set theory to $\mathsf{ZF}$ (= set theory without the axiom of choice), there is a genuine difference between "all well-orderable languages" and "all languages." It's worth noting first of all that model theory, even over $\mathsf{ZF}$ alone, does not confine itself to well-orderable languages; the situation is simply that there are theorems which only apply to well-orderable languages, just as (even in $\mathsf{ZFC}$) there are theorems only applicable to finite languages, or to countable languages, or to languages satisfying some other condition.

• Interestingly, over $\mathsf{ZF}$ most of the natural uncountable languages are not provably well-orderable - e.g. the language of nonstandard analysis above. There are still examples which are important, but they tend to be more technical.

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And as usual, in my opinion the answer to

if so, what do we gain by a study of such languages?

is "The resulting theory is interesting." There are certainly (e.g. via nonstandard analysis) uncountable languages with applications outside pure model theory, but I think that's secondary: part of the takeaway of model theory is exactly the rich and surprising behavior of structures in "large" languages.