Abstract: 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.