The Axiomatic System of Difference Generation Δ and Its Universal Connectivity — A Proposal for a Generative Framework in the Foundations of Mathematics —

We propose an axiomatic system of difference generation Δ, addressing fundamental questions in mathematics and ontology. While conventional mathematical systems presuppose the existence of distinguished objects or sets, this approach focuses on how such distinctions are generated.The framework of difference generation Δ operates as the generator of difference between entities, describing the conditions for their formation through generative operations without presupposing the existence of mathematical objects. This perspective invites a reconsideration of the foundational assumptions behind set theory, type theory, and category theory.We formulate the basic structure of the axiomatic system through the definition of difference generation Δ and connection operation Λ, which connects entities within this framework. The system includes an extended set of axioms that systematically describe its structure. We also explore potential connections with modern mathematics (category theory), physics (particle interactions), and information theory (information entropy). In the context of information theory, the difference generation Δ can be defined as the information quantity I:I(A, B) := Δ(A, B).The generative framework proposed here aims to bridge ontology and the foundations of mathematics by describing the phase preceding the establishment of fundamental mathematical objects. This approach may offer a basis for future interdisciplinary research and theoretical development.