Frames and Quantities: A Construction Approach to Set Theory (Revised Third Version)

This paper proposes a dual-attribute construction method for set theory based on "rules (Frames) + carriers (Quantities)", which differs from the underlying assumption of classical ZFC axiomatic set theory that defines a set as a collection of elements. This paper argues that rules are not external axiomatic constraints attached to sets, but intrinsic attributes of sets. Starting from this standpoint, we redefine fundamental concepts including sets, empty sets, identity, union, partition, filtering and mapping, and construct an axiomatic system consisting of nine axioms. We further analyze the correlations between this system and the three postulates of formal logic (the Law of Identity, the Law of Contradiction, the Law of Excluded Middle): the three universal propositions of absolute validity are transformed into locally valid laws "conditional on a given Frame". This paper does not presuppose superiority or inferiority relative to ZFC, and presents it as a parallel construction path. It demonstrates stronger intuitive explanatory power in scenarios such as coexistence across Frames, fuzzy boundaries, and non-unique empty sets. Partial directions (special axioms for infinite cardinals, comprehensive comparison with non-classical logic) are not fully elaborated in this manuscript and reserved for future extensions.