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Chapter 1: The Fundamental Failure of ZFC and Statement of RH

1.1 The Incompleteness of Current Mathematics​

Theorem 1.1 (Gödel): ZFC cannot prove its own consistency.

Critical Observation: All attempts to prove RH within ZFC are doomed because ZFC is fundamentally incomplete. It cannot even validate itself, yet mathematicians pretend it's a solid foundation.

1.2 Statement of the Riemann Hypothesis​

Definition: For Re(s) > 1: ζ(s)=∑n=1∞1ns=∏p prime11−p−s\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}

The Riemann Hypothesis: All non-trivial zeros of ζ(s) have Re(s) = 1/2.

1.3 Why ZFC Cannot Prove RH​

Theorem 1.2: RH is unprovable in ZFC because it's a statement about arithmetic self-consistency, which Gödel showed is impossible within the system.

Proof: RH encodes arithmetic's self-reference through the Euler product. ZFC excludes self-reference to avoid paradoxes, making RH unprovable. ∎

Continue to Chapter 2: Mathematical Prerequisites