As far as I can tell, that is what most mathematicians use most of the time.

Of course, in ordinary mathematics, there’s no formal system at work, so it doesn’t really make sense to reconstruct the argument one way rather than the other, so I’m not sure if it’s quite right to compare this style of proof with the one at work here.

Although, people often call reductio an indirect proof, and in natural deduction systems that’s the only standard way to prove a negative, so maybe there’s something to this. (Perhaps it only counts as indirect if it also uses a double-negation step?)

The example from non-standard analysis feels much more similar to me. One proves that statements of a certain sort are true in the reals iff in the hyperreals, and then proves it for the hyperreals, so it must be true in the reals. Similarly, one proves that statements of a certain sort are true in all universes of sets iff they are true in a universe with CH, and then proves it for a universe with CH, so it must be true in all universes of sets. Viewed in this metatheoretic way, these proofs aren’t even really indirect - they really do give a proof of the result, though perhaps the phrasing of the result is different than one expected.

Another nice example is the metatheorem of non-standard analysis that says that proofs of the right type using infinitesimals can be converted into theorems of ordinary analysis using conventional deltas and epsilons.

A Horn sentence is logically equivalent to a conjunction of conditional statements having one of the following forms:

1. (A1/\…/\An) -> B
2. (A1/\…/\An) -> False
3. True -> B

It was Horn who established a sufficient condition for an elementary class to be closed under direct product. Chang and Morel showed that this is not a necessary condition. Appel showed that, if condition is restricted to identity theory (first order predicate calculus with equality but no other relation symbols or operation symbols), then Horn’s condition is necessary and sufficient.

Horn Sentences in Identity Theory
K. I. Appel
The Journal of Symbolic Logic, Vol. 24, No. 4 (Dec., 1959), pp. 306-310

I’m pretty sure that the Chang of Chang and Morel is thew same as that of Chang and Keisler’s Model Theory.

Which reminds me of the drunk who phoned the TV network when Connie Chung was anchoring a broadcast after a minor abrasion to her face from a household accident.

“What,” he asked, “happened to Connie Chin’s chung?”

(I assume that if they could actually construct the proof without CH, they would just present that)