Equality of Real Numbers as a Consistency Statement

This paper investigates how the equality of two explicitly defined computable real numbers can encode the consistency of a formal axiomatic theory. Starting from the binary expansion of an irrational number, the construction defines real numbers whose digits depend on whether a formal proof of contradiction has been found in Peano Arithmetic (PA) or Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). Their equality is thereby connected to a consistency statement. The central result establishes, within the stated formal framework, an equivalence between the equality of the constructed real numbers and the consistency of ZFC. Gödel's Second Incompleteness Theorem then provides a striking consequence: assuming ZFC is consistent, the two numbers are equal, yet their equality cannot be proved within ZFC itself. The construction illustrates the distinction between mathematical truth and formal provability through a concrete numerical example. It shows how a seemingly elementary question about equality can carry the full logical difficulty of a system's ability to establish its own consistency. Beyond its mathematical content, the paper raises a philosophical question about the relationship between truth, proof, and mathematical existence: What constitutes truth when it cannot be established? The paper distinguishes the mathematical equivalence from the further requirements of a fully machine-checked formalization.