Who Put The “Back” In Back-And-Forth?

Abstract

Logicians of a certain age can remember when they were first allowed to savor Cantor’s sweet theorem characterizing the rationals among ordered sets as the unique countable densely ordered set without endpoints. For most the pleasure was heightened when they were shown the elegant back-and-forth argument that makes the proof of Cantor’s theorem perfectly transparent. This method of establishing isomorphism is now so widely known that its invocation no longer begins demonstrations, it ends them. Provers tell us: “By the usual back-and-forth argument so-and-so is isomorphic to such-and-such”, and then they move on to other matters. Logicians, in particular model-theorists, deserve a great deal of credit for making back-and-forth a mathematical cliché.