Philosophia Mathematica 1 (1):3-23 (1993)
is the second-order theory of the part-whole relation. It can express such hypotheses about the size of Reality as that there are inaccessibly many atoms. Take a non-empty class to have exactly its non-empty subclasses as parts; hence, its singleton subclasses as atomic parts. Then standard set theory becomes the theory of the member-singleton function—better, the theory of all singleton functions—within the framework of megethology. Given inaccessibly many atoms and a specification of which atoms are urelements, a singleton function exists, unique up to isomorphism. (This article is partly abridged from my Parts of Classes, partly a sequel.).
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Identity, Indiscernibility, and Ante Rem Structuralism: The Tale of I and –I.Stewart Shapiro - 2007 - Philosophia Mathematica 16 (3):285-309.
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