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 5 - Aoo-Definable functions
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
Calculable functions
An Aoo-function is an Aoo-formula of type u, ui, …, these are called one-, two-, …place Aoo-functions. If ρ is a one-place Aoo-function and if v is a numeral then (ρv) is a numerical term and determines a unique numeral called the value of the function ρ for the argument v. Similarly if ρ is a two-place Aoo-function and if v and K are numerals then ((ρv)K) is a numerical term and so determines a unique numeral called the value of the function ρ for the arguments v and K, in that order. Similarly for many-place functions. A calculable function of natural numbers is a rule or set of rules such that given the ordered argument set we can by following the rules effectively find a natural number called the value of the function for that argument set, the value must be unique and a given function always requires the same number of arguments. Thus the Aoo-functions are a particular kind of calculable function in that the rules for finding the value given the argument set is of a particular kind. For example a set of instructions to find the value of a function of natural numbers might be: replace (S(Sv)) by v as long as possible, this would replace an even number by 0 and an odd number by (SO).
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- Information
- Mathematical Logic with Special Reference to the Natural Numbers , pp. 232 - 277Publisher: Cambridge University PressPrint publication year: 1972