theorem A.29 Second incompleteness theorem

open in the book · appendices/A-long-proofs.tex:2510 · p. 2804

Rests on

depends_on equation 3.89 eq:goedel-sentence ¶
depends_on proposition A.28 prop:app-inc-derivability ¶
- depends_on equation A.129 eq:app-inc-prov ¶
- depends_on theorem A.17 Representability ¶
  - depends_on definition A.16 Representability ¶
    - depends_on definition 3.80 Formal system ¶
  - depends_on definition 3.93 Computable function, decidable set ¶
    - depends_on definition 3.92 Turing machine ¶
  - depends_on lemma A.20 $\Sigma_{1}$-completeness ¶
    - depends_on definition A.18 $\Delta_{0}$ and $\Sigma_{1}$ ¶
      - depends_on equation A.119 eq:app-inc-order ¶
    - depends_on lemma A.19 Numerals behave ¶
      - depends_on definition 3.83 Peano arithmetic ¶
      - depends_on equation A.118 eq:app-inc-Q-successor ¶
      - depends_on equation A.119 eq:app-inc-order ¶ ↺
      - proves proof app:A-long-proofs@proof-13 ¶
    - depends_on proposition 3.22 Algebra of propositions ¶
      - depends_on definition 3.17 Logical equivalence ¶
      - depends_on definition 3.6 Truth table ¶
      - proves proof ch:01-logic-sets@proof-4 ¶
    - proves proof app:A-long-proofs@proof-14 ¶
  - depends_on theorem A.25 thm:app-inc-primrec ¶
    - depends_on definition A.16 Representability ¶ ↺
    - depends_on lemma A.23 Sequence lemma ¶
      - depends_on definition A.22 Gödel's $\beta$ ¶
      - depends_on lemma A.21 Chinese remainder theorem ¶
      - proves proof app:A-long-proofs@proof-16 ¶
    - depends_on lemma A.24 lem:app-inc-beta-delta0 ¶
      - depends_on definition A.22 Gödel's $\beta$ ¶ ↺
      - depends_on definition A.18 $\Delta_{0}$ and $\Sigma_{1}$ ¶ ↺
      - depends_on lemma A.19 Numerals behave ¶ ↺
      - proves proof app:A-long-proofs@proof-17 ¶
    - depends_on lemma A.19 Numerals behave ¶ ↺
    - depends_on lemma A.20 $\Sigma_{1}$-completeness ¶ ↺
    - proves proof app:A-long-proofs@proof-18 ¶
  - proves proof app:A-long-proofs@proof-19 ¶
- proves proof app:A-long-proofs@proof-22 ¶
depends_on theorem 3.87 Gödel, 1931 ¶
- depends_on definition 3.82 Consistency, completeness, soundness ¶
  - depends_on definition 3.80 Formal system ¶ ↺
  - depends_on definition 3.7 Negation ¶
- depends_on definition 3.81 Effective axiomatization ¶
  - depends_on definition 3.80 Formal system ¶ ↺
- depends_on equation 3.86 eq:provability-predicate ¶
- depends_on lemma 3.86 Diagonal lemma ¶
  - depends_on definition 3.83 Peano arithmetic ¶ ↺
  - depends_on definition 3.80 Formal system ¶ ↺
  - proves proof ch:01-logic-sets@prooflink-3 ¶
- proves proof ch:01-logic-sets@proof-20 ¶
proves proof app:A-long-proofs@proof-23 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:goedel-sentence	declared	appendices/A-long-proofs.tex:2513
`depends_on`	→	prop:app-inc-derivability	declared	appendices/A-long-proofs.tex:2513
`depends_on`	→	Gödel, 1931	declared	appendices/A-long-proofs.tex:2513
`proves`	←	app:A-long-proofs@proof-23	declared	appendices/A-long-proofs.tex:2516