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Chapter 3: The Self-Consistency Principle

3.1 The Fundamental Axiom​

Axiom (Self-Consistency): Mathematical structures exist if and only if they are self-consistent.

Critical Contrast with ZFC:

  • ZFC assumes consistency but cannot prove it (Gödel)
  • ZFC avoids self-reference, creating incompleteness
  • Our framework builds on self-consistency as primary

Definition (Consistency Operator): C(M)={Mif M is self-consistent∅otherwise\mathcal{C}(M) = \begin{cases} M & \text{if } M \text{ is self-consistent} \\ \emptyset & \text{otherwise} \end{cases}

3.2 Key Theorems​

Theorem 3.1: C\mathcal{C} is idempotent: C(C(M))=C(M)\mathcal{C}(\mathcal{C}(M)) = \mathcal{C}(M).

Proof: If M is consistent, C(M)=M\mathcal{C}(M) = M, so C(C(M))=C(M)\mathcal{C}(\mathcal{C}(M)) = \mathcal{C}(M). ∎

Theorem 3.2: Arithmetic ℕ is a fixed point: C(N)=N\mathcal{C}(\mathbb{N}) = \mathbb{N}.

Proof: Arithmetic has no contradictions, satisfies its defining properties. ∎

Theorem 3.3 (Meta-Consistency): The consistency principle itself is self-consistent.

Proof: C("consistency principle")="consistency principle"\mathcal{C}(\text{"consistency principle"}) = \text{"consistency principle"}, creating a necessary fixed point. Unlike ZFC which cannot address its own consistency, our framework is self-validating. ∎

Continue to Chapter 4: Arithmetic as a Self-Referential System