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Chapter 4: Arithmetic as a Self-Referential System

4.1 The Fundamental Theorem of Arithmetic​

Theorem 4.1 (Fundamental Theorem): Every n > 1 has unique prime factorization.

Critical Insight: This theorem reveals arithmetic's inherent self-reference—numbers defined by other numbers.

4.2 Self-Reference vs ZFC's Failures​

ZFC's Circular Disaster:

  • Natural numbers defined using... natural numbers (successor function)
  • Peano axioms assume what they claim to define
  • Induction principle requires the very structure it establishes

Our Framework: Self-reference is explicit via ψ = ψ(ψ).

4.3 Key Theorems​

Theorem 4.2: Unique factorization ⟺ arithmetic self-consistency.

Proof: Non-unique factorization → contradictory divisibility → inconsistent arithmetic. ∎

Theorem 4.3: The Euler product encodes arithmetic self-reference: ∑n=1∞1ns=∏p11−p−s\sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p} \frac{1}{1-p^{-s}}

Proof: Left = additive view, Right = multiplicative view. Their equality IS arithmetic recognizing itself. ∎

Theorem 4.4: Zeros of ζ(s) are points where arithmetic's self-recognition achieves perfect balance.

Proof: At zeros, all arithmetic aspects cancel perfectly—the ultimate self-referential statement. ∎

Continue to Chapter 5: The Zeta Function as Arithmetic Mirror