The varieties of mathematical basis for formalizing linguistic theories are more diverse than is commonly realized. For example, the later work of Zellig Harris might well suggest a formalization in terms of CATE-

We address certain recent suggestions that the existence of infinitely many grammatical expressions in human languages (the infinitude claim) is a universal of human language. We examine the arguments given for the infinitude claim, and show that they tacitly depend on the unwarranted assumption that the only way to represent the structural properties of a language is by means of a generative grammar with a recursive rule system. We explore some of the reasons why linguists have been so willing to accept language infinitude despite its inadequate support and its paucity of linguistic consequences. We suggest that the infinitude claim is motivated chiefly by an inadvisable adherence to the notion that languages are sets. It is not motivated by considerations of the creative aspect of language use, or opposition to associationist psychology, or the putative universality of iterable linguistic structure such as recursive embedding or unbounded coordination (which are in any case probably not universal). 1 Infinitude as a linguistic universal In a number of recent works, linguists have portrayed the infinitude of sentences in human languages as an established linguistic universal. Lasnik (2000) asserts, in the opening chapter of a textbook based on transcriptions of a series of introductory syntax lectures: