Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- Acknowledgements
- Greek alphabet
- 1 Untyped lambda calculus
- 2 Simply typed lambda calculus
- 3 Second order typed lambda calculus
- 4 Types dependent on types
- 5 Types dependent on terms
- 6 The Calculus of Constructions
- 7 The encoding of logical notions in λC
- 8 Definitions
- 9 Extension of λC with definitions
- 10 Rules and properties of λD
- 11 Flag-style natural deduction in λD
- 12 Mathematics in λD: a first attempt
- 13 Sets and subsets
- 14 Numbers and arithmetic in λD
- 15 An elaborated example
- 16 Further perspectives
- Appendix A Logic in λD
- Appendix B Arithmetical axioms, definitions and lemmas
- Appendix C Two complete example proofs in λD
- Appendix D Derivation rules for λD
- References
- Index of names
- Index of definitions
- Index of symbols
- Index of subjects
14 - Numbers and arithmetic in λD
Published online by Cambridge University Press: 05 November 2014
- Frontmatter
- Contents
- Foreword
- Preface
- Acknowledgements
- Greek alphabet
- 1 Untyped lambda calculus
- 2 Simply typed lambda calculus
- 3 Second order typed lambda calculus
- 4 Types dependent on types
- 5 Types dependent on terms
- 6 The Calculus of Constructions
- 7 The encoding of logical notions in λC
- 8 Definitions
- 9 Extension of λC with definitions
- 10 Rules and properties of λD
- 11 Flag-style natural deduction in λD
- 12 Mathematics in λD: a first attempt
- 13 Sets and subsets
- 14 Numbers and arithmetic in λD
- 15 An elaborated example
- 16 Further perspectives
- Appendix A Logic in λD
- Appendix B Arithmetical axioms, definitions and lemmas
- Appendix C Two complete example proofs in λD
- Appendix D Derivation rules for λD
- References
- Index of names
- Index of definitions
- Index of symbols
- Index of subjects
Summary
The Peano axioms for natural numbers
In the previous chapters we have become acquainted with the use of λD for doing mathematics, by selecting a few examples and investigating the issues that we came across.
Let's now make a fresh start by thoroughly exploring the most fundamental entities in mathematics: natural and integer numbers. This will not be easy, since in the process of development we have to pretend that we ‘know nothing’ about subjects we are so familiar with. As a consequence, we have to build up our knowledge from scratch, which may seem cumbersome, but it is also quite interesting, since we are obliged to scrutinise the foundations of mathematics.
In the present section, we start with the basis: natural numbers. Integers will be the main topic of following sections.
In Chapter 1 we saw how natural numbers, and operations on naturals such as addition and multiplication, can be coded in untyped lambda calculus, as so-called Church numerals (see Exercise 1.10). There also exist encodings of these notions in typed lambda calculi: in the chapter about λ2 we have discussed the so-called polymorphic Church numerals; see, for example, Section 3.8 and Exercise 3.13. (For Church numerals in λ→: see Section 2.14.)
Therefore, it would be a type-theoretically justified choice to introduce the natural numbers in this manner. This can be done by writing down the appropriate definitions, since λ2 is a subsystem of λD.
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- Chapter
- Information
- Type Theory and Formal ProofAn Introduction, pp. 305 - 348Publisher: Cambridge University PressPrint publication year: 2014