What is the Set
Yao Cong-jun · Dianzi Ke-Ji Daxue xuebao. She-ke ban · 2010
The Cantorian definition of sets,which are commonly used to justify ZFC,has two main readings for sets: the limitation-of-size conception and iterative conception.The limitation-of-size conception understands the unity or the thingness of a collection as related to its size,but it offers no justification for either the foundation axiom or the power-set axiom.The iterative idea of set imposes a temporal restriction on the set-formation,and it justifies separation axiom and power set axiom,but it fails to offer an explanation for replacement axiom.The most natural way to do it would be to impose a limitation-of-size on the iterative,which can explain replacement,axiom but fails to justify power set axiom.Structural conception considers set as a pattern of unfolding a possible structure,so the identity of sets should be given by the identity of their analytical patterns.Under this conception there is no reason to limit the possible structure to the well-founded one.