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Is Cantorian set theory an iterative conception of set? (English) Zbl 0747.03003

The author argues that nothing in Cantor’s publications or letters justifies the claim, attributed by the author to Boolos, Parsons, and Wang, that Cantor’s notion of set involves the iterative conception of set. In Cantor’s principal work, the 1883 Grundlagen paper and the 1895-1897 Beiträge, as well as all his other papers, the author sees only the naive picture of a set, either as the extension of a concept or as a more nebulous ‘collection’ of objects. In some of his later correspondence with Dedekind, Jourdain, and Young, Cantor proposed a limitation of size principle and suggested that he operated according to this principle as early as the Grundlagen paper. The author is skeptical of this suggestion and believes that Cantor’s “memory failed him”. (The reviewer suspects that there also might have been some ‘face- saving’ involved; Cantor always maintained that the set-theoretic paradoxes could not be derived on the basis of his conception of set.) In any case, the limitation of size principle does not necessarily require the iterative conception of set (although the axiomatic development of the iterative conception does make use of a limitation of size principle). Thus, in the author’s view, there is no trace of the iterative conception in Cantor’s ideas.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03-03 History of mathematical logic and foundations
01A55 History of mathematics in the 19th century
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