Abstract

After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt (1930) to secure ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair.

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