Sets and Properties with Indeterminate Identity
Terence D. Parsons · Oxford University Press eBooks · 2000
This chapter discusses sets and properties with indeterminate identity. Sets are things whose identities are defined in terms of their members. They are identical if they both determinately have and determinately lack the same members. They are distinct if one of them determinately has a member that the other determinately lacks, and otherwise it is indeterminate whether they are identical. The discussion distinguishes between worldly sets and conceptual sets. The former but not the latter obey the DDiff principle for set membership. By construing worldly properties as extensional, they can be identified with the sets under discussion, and the resulting theory can be extended to a transfinite hierarchy of sets, yielding an indeterminate version of Zermelo–Fraenkel (ZF) set theory. Meanwhile, relations are definable as sets of ordered pairs, using a refinement of the usual definition of ordered pairs.