Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles
Article
Formal Antecedent
Foundations and Logic
Citation
Dan E. Willard. (2001). Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles. Journal of Symbolic Logic. 66(2). pp. 536–596.
Why this reference is included
Willard’s 2001 Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles, published in Journal of Symbolic Logic, is one of the program’s working technical references. Cited across Book I (Categorical Foundations), Part 18, Chapter The Self-Hosting Landscape; Book I (Categorical Foundations), Part 18, Chapter The Enrichment Frontier — the central framing is “Willard constructed formal arithmetic theories that can prove their own consistency”.
Cited in
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Book I — Categorical Foundations Part 18Chapter The Self-Hosting Landscape
Willard constructed formal arithmetic theories that can prove their own consistency
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Book I — Categorical Foundations Part 18Chapter The Enrichment Frontier
Precedent. Two programs have approached E_3 from different directions, each achieving it partially: Willard constructs self-verifying axiom systems — theories that prove their own consistency without violating G\"odel's second incompleteness theorem
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Book I — Categorical Foundations Part 18Chapter The Enrichment Frontier
Willard : self-verification at sub-PA strength
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Book I — Categorical Foundations Part 18Chapter The Enrichment Frontier
_0 → E_1 & Altenkirch–Kaposi 2016 ; Bocquet–Kaposi–Sattler 2023 & Non-Boolean, constructive adaptation to four-valued Ω_τ & Earned topos E_τ (I.D59) , E_1 → E_2 & Joyal arithmetic universes ; Abel graded modal DTT & Linear DTT not yet complete; no internal cut-elimination & 5 diagonal discipline; three-grade semiring , E_2 → E_3 & Willard 2001 (weak); Girard TX (fragments) & No full system at CIC-level proof-theoretic strength & *-autonomous proof theory via 5 (I.T39) , Book III proposes to attempt something that has not been done before