Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Formal systems
- Chapter 2 Propositional calculi
- Chapter 3 Predicate calculi
- Chapter 4 A complete, decidable arithmetic. The system Aoo
- Chapter 5 Aoo-Definable functions
- Chapter 6 A complete, undecidable arithmetic. The system Ao
- Chapter 7 Ao-Definable functions. Recursive function theory
- Chapter 8 An incomplete undecidable arithmetic. The system A
- Chapter 9 A-Definable sets of lattice points
- Chapter 10 Induction
- Chapter 11 Extensions of the system AI
- Chapter 12 Models
- Epilogue
- Glossary of special symbols
- Note on references
- References
- Index
Chapter 11 - Extensions of the system AI
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Formal systems
- Chapter 2 Propositional calculi
- Chapter 3 Predicate calculi
- Chapter 4 A complete, decidable arithmetic. The system Aoo
- Chapter 5 Aoo-Definable functions
- Chapter 6 A complete, undecidable arithmetic. The system Ao
- Chapter 7 Ao-Definable functions. Recursive function theory
- Chapter 8 An incomplete undecidable arithmetic. The system A
- Chapter 9 A-Definable sets of lattice points
- Chapter 10 Induction
- Chapter 11 Extensions of the system AI
- Chapter 12 Models
- Epilogue
- Glossary of special symbols
- Note on references
- References
- Index
Summary
The system A′
We have seen that the system AI is incomplete and that any extension of it which remains a formal system will also be incomplete. We could add, as extra axioms, some A-true but AI-unprovable statements so as to obtain more A-true statements as theorems in the resulting extended system. But as long as we have a formal system it will still be incomplete and an irresolvable statement can be constructed on the same lines as before.
We can do this programme in a systematic manner as follows: We have an effective method for constructing an irresolvable ℒ-true ℒ-statement in a formal system ℒ which contains recursive number theory and negation. Call this ℒ-statement G{ℒ}. We first form G{AI} and then the system G′ which consists of the system AI with the extra axiom G{AI}; having formed G(λ) we construct G(Sλ) by adding the extra axiom G{G(λ)}. Having formed the systems AI, G′,…, G(λ),… we then form the system G* as the union of all the systems AI, G′,…, i.e. the system AI with all the extra axioms we added in forming the systems G′,…; this system again will be formal, hence we can form G{G*} and the system G*′ which is the system G* plus the extra axiom G{G*}. So we can proceed through the constructive ordinals. But we shall have to stop before we come to the end of the constructive ordinals, otherwise we shall cease to have a formal system.
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- Mathematical Logic with Special Reference to the Natural Numbers , pp. 511 - 562Publisher: Cambridge University PressPrint publication year: 1972