DEFINABILITY IN DISREPUTE

While what Alfred Tarski labeled his “semantic conception of truth” has been much discussed, one topic that has not received all the attention it deserves is his choice of that label. It is this comparatively neglected aspect of Tarski's conception that I wish to address here. But first a word about the situation prior to Tarski.

I begin with a result I learned as a fifteen-year-old student in a summer mathematics program for high-school students run by the late Arnold Ross: the theorem that every natural number is interesting. The proof is by contradiction. Suppose that not every natural number is interesting. Then the set of uninteresting natural numbers is non-empty. So by the well-ordering property of the natural numbers, it must have a smallest element n. But if n is the smallest uninteresting natural number, then n is interesting for that very reason. Thus we have a contradiction, establishing that our original hypothesis was false, and that every natural number is interesting after all. But, of course, some numbers do appear completely uninteresting to most of us. I suppose a so-called dialethist might claim that here we have yet another example of a true contradiction, but the more usual reaction to this bit of adolescent mathematical humor is that “interesting” is too vague or ambiguous, too subjective or relative, a concept to be admissible in mathematical reasoning. And when Alfred Tarski was beginning his mathematical career, most mathematicians held essentially the same opinion about the concept of truth.