Abstract

This paper introduces the Co-Equal Structure Thesis (CEST), the proposal that the identity of any infinite set S is given by the ordered pair Φ(S) = ⟨card(S), ord(S)⟩. After showing that Φ is definable within standard ZFC and that ZFC + Φ constitutes a conservative definitional extension, the paper argues that cardinality alone is operationally insufficient for distinguishing structurally divergent sets. Ordinal-sensitive operations—such as ordinal multiplication—produce distinct order-types that cardinal equivalence collapses. CEST is offered as a conceptual remedy: a dual-invariant identity criterion integrating structural adequacy with extensional economy. The paper concludes with implications for foundations, suggesting the possibility of an Axiom of Structural Identity (ASI), and outlines potential directions for further logical and philosophical development.