Now That’s Indirect

In Chang and Keisler’s Model Theory, they give one of the most indirect proofs I’ve ever seen. They prove a theorem assuming the continuum hypothesis, and then they prove a metatheorem that any theorem of that particular form proven using the continuum hypothesis must also have a different proof that does not use the continuum hypothesis. They don’t actually exhibit this proof — they just prove that it exists. (The theorem shows that a theory preserves reduced products if and only if it can be defined entirely in terms of Horn sentences.)