Extension Cardinals and Pre-type Structures: A New Foundation for Mathematics

Traditional foundations of mathematics are centered on ZFC set theory. However, the independence of the Continuum Hypothesis reveals that the concept of cardinals overlooks mathematical structures, and infinite-dimensional objects lack a rigorous status. This paper proposes an extension of cardinal theory by introducing a more fundamental notion than sets: the \emph{pre-type structure}. Every mathematical object is generated from basic extension cardinals through an \emph{extension relation}, and its essence is characterized by structural invariants and modal properties. Concurrently, we realize modal logic as a geometric structure within a \emph{moduli space}, endowing necessity and possibility with an algebraic-geometric semantics. Based on this, we establish a new axiom system, \textbf{EZFC}, which unifies set theory, modal logic, and moduli spaces. This framework transforms the Continuum Hypothesis into a structure-sensitive, multi-dimensional issue, providing a rigorous foundation for infinite-dimensiona