More than a generation of linguistics professors, myself included, have harped at our students about the importance of human language building an infinite edifice from finite materials. The hoopla more or less reached its zenith in 2002, when Hauser, Chomsky and Fitch made much of the human capacity for recursion in language--indeed they proposed that it was the main thing which separated us from the other apes. But beyond the conventional wisdom, as usual, truly interesting things were being discovered.
Over the past ten or more years, Dan Everett, for one, slowly convinced the linguistics community that the Pirahã language of the Amazon in fact has no kind of recursive or iterative grammatical structures. But other less highly advertised cases had long been lurking, among them Dyirbal and Warlbiri. What is linguistic theory to make of finite human languages? Why would there be any, when we long thought that "infinitude" was a necessary property of a possible human language?
Although some have suggested otherwise, I think by now we all must admit that these descriptions are indeed correct, that there are really finite human languages, and that this is within the scope of possibility. It's OK--Kornai shows that this is no cause for alarm. The argument rests on the important point that even in the infinite languages, such as English, there is a steadily vanishing probability of a sentence being produced as the length increases. This probability distribution over the length of sentences shows that 99.9% of everything English speakers actually say is communicated in relatively short simple sentences. On another note, whatever the infinitude of English is good for, it is not good for saying anything beyond what Pirahã speakers can say--their expressive power is the same, so long as a Pirahã speaker is allowed to use multiple sentences to say what might be said using one English sentence. In this respect, the information capacity of real English is similar to that of Pirahã. In fact, a mathematical argument shows that a more complicated finite language could easily outstrip the information capacity of infinitary natural languages as they actually are spoken.
One conclusion to draw is that all the hub-bub about recursion and infinity in grammar being the essence of the human condition is seriously misguided. Indeed, the evolutionary pressure, whatever it is, that leads most human languages to have infinitary sentences is a bit of a mystery, since it appears to provide little discernible advantage. Except perhaps for the inherent value of nifty tales like "This is the House that Jack Built." For fun, here is its final sentence: