Formal Analysis of the Instability of the Empty Set and the Necessity of Existence

This document proposes an answer to the metaphysical question "Why is there something rather than nothing?" through the mathematical framework of the Zermelo-Fraenkel (ZF) axiom system. We prove that "Absolute Nothingness," represented by the empty set ($\emptyset$), is **Structurally Unstable** under systems containing the logic of Identity or Recursion. The result is **The Instability of the Void Theorem**, which indicates that the existence of structure (Existence) is not accidental, but a specific **Logical Necessity** inevitable within a complete closed system.