A Reconsideration of the Definiteness and Extensibility of the Infinite Totalities

From the Viewpoint of Begriff and Umfang

Abstract

According to intuitionism (or more broadly put, constructivism), infinite totalities such as the totality of natural numbers should be conceived of as potential infinites ; they enlarge their extensions indefinitely as our mathematical investigations proceed. The classical conception, in contrast, takes the totalities to have completely definite ranges of members and treats each of the totalities as a full-fledged object on its own. The author reconsiders the problem by concentrating on how we introduce infinite totalities as domains of quantification into our discourse in general.The conclusion is that some sort of intensional principle (something like Fregean Begriff) is constinutive of our conception of the domains ; the quantifier involved ranges over an infinite domain that includes not only actual objects, but also possible objects that are (partially) specified by the principle. Although this observation would not necessarily vindicate intuitionism, it certainly throws doubts on the classical conception of domains as sets and of the totality of sets as the completely definite universe.