A Sub-aleph-zero (Sub-ℵ0) Axiom System

We propose a foundational extension of axiomatic set theory — a Sub-aleph-zero (Sub-ℵ0) Axiom System — in which infinity is not treated as atomic at ℵ0, but instead admits an internal, stratified hierarchy of countably infinite cardinalities strictly below ℵ0. In classical ZFC, ℵ0 functions as the minimal infinite cardinal and does not allow internal cardinal refinement: every infinite subset of N has the same cardinality, and no strictly intermediate infinite sizes exist below ℵ0. Our work reexamines this structural feature and develops a coherent formal framework in which subℵ0 magnitudes coexist consistently with standard infinite cardinals. The proposed system introduces additional axioms that govern internal stratification, together with a controlled extension of cardinal arithmetic that allows the definition of a transfinite logarithmic operator acting below ℵ0. This operator formalizes an inverse relationship to exponentiation within the sub-infinite hierarchy and provides a structural mechanism for distinguishing refined levels of countable infinity. We analyze the interaction of this framework with classical set-theoretic principles, discuss relative consistency considerations, and outline model-theoretic interpretations inspired by nonstandard and non-well-founded approaches. The resulting theory preserves the global architecture of higher cardinalities while enriching the first infinite level with internal structural depth.