The Indecidability of the Existence of Weakly Inaccessible Cardinals in ZFC: A Gödel-Inspired Logical Approach

<p>This paper explores the undecidability of the existence of weakly inaccessible cardinals<br>in the von Neumann universe under Zermelo-Fraenkel set theory with the Axiom of Choice<br>(ZFC). Through a novel logical framework inspired by Gödel’s incompleteness theorems,<br>we demonstrate that the statement “there exists a weakly inaccessible cardinal” is neither<br>provable nor disprovable in ZFC, assuming ZFC is consistent. The framework introduces<br>intuitive concepts such as the intrinsic distinctness of “super-large” objects, a “functional<br>escape” from the properties of smaller objects, and the limits of formal systems. These<br>ideas are formalized to show that weakly inaccessible cardinals transcend ZFC’s constructive<br>capabilities, aligning with Gödelian incompleteness. The paper provides technical<br>details on forcing and the constructible universe L, clarifies the original contribution of<br>the logical approach, and contextualizes weakly inaccessible cardinals among other large<br>cardinals. Implications for set theory and future research directions are discussed.</p>