theorem A.17 Representability

open in the book · appendices/A-long-proofs.tex:2006 · p. 2799

Rests on

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 axiom 3.33 Peano axioms ¶
    - depends_on axiom 3.34 Principle of induction ¶
    - depends_on definition 3.80 Formal system ¶ ↺
  - 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 3.12 Biconditional ¶
    - depends_on definition 3.14 Tautology ¶
  - 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-15 ¶
  - 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 ¶

Supports

depends_on proposition A.28 prop:app-inc-derivability ¶
- depends_on theorem A.29 Second incompleteness theorem ¶
depends_on theorem A.30 Church, Turing ¶
depends_on theorem A.26 Diagonal lemma ¶
- depends_on theorem A.27 Rosser ¶
depends_on theorem A.27 Rosser ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Representability	declared	appendices/A-long-proofs.tex:2012
`depends_on`	→	Computable function, decidable set	declared	appendices/A-long-proofs.tex:2012
`depends_on`	→	$\Sigma_{1}$-completeness	declared	appendices/A-long-proofs.tex:2012
`depends_on`	→	thm:app-inc-primrec	declared	appendices/A-long-proofs.tex:2012
`depends_on`	←	prop:app-inc-derivability	declared	appendices/A-long-proofs.tex:2489
`depends_on`	←	Church, Turing	declared	appendices/A-long-proofs.tex:2581
`depends_on`	←	Diagonal lemma	declared	appendices/A-long-proofs.tex:2356
`depends_on`	←	Rosser	declared	appendices/A-long-proofs.tex:2428
`proves`	←	app:A-long-proofs@proof-19	declared	appendices/A-long-proofs.tex:2288