Abstract
The Natural Deduction System of First-Order Logic studied in Chapters 1 and 2 supplies all the general proof techniques needed for mathematics and any other deductive enterprise. However, different fields of study may have specialized proof strategies tailored to their subject matter. Natural number arithmetic uses Proof by Mathematical Induction, a technique whose importance it gets hard to overestimate, because it’s used in all areas of mathematics. This chapter explores it in its home setting, along with the closely related Definition by Recursion. Induction is also explored in two other directions, looking at how to determine closed formulas for recurrence relations, and showing how induction is used in some non-numerical settings (theory of strings, well-formed formulas). We then explore how the theory of arithmetic can be developed axiomatically from the Peano Postulates. Finally, the chapter concludes by looking at divisibility and the greatest common divisor, something that gives us a basis for discussing modular arithmetic in Chapter 6.
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Notes
- 1.
Invented by mathematician and Fibonacci aficionado E. Lucas in 1883, versions of this game and a discussion of its history can be found on numerous web sites.
- 2.
For a discussion of this theorem, see Leon Henkin’s April 1960 article On Mathematical Induction in The American Mathematical Monthly.
- 3.
- 4.
On axiomatizing the theory of strings, see the article String Theory by John Corcoran et al. in the December, 1974 issue of The Journal of Symbolic Logic.
- 5.
This is treated in A Simple Decision Procedure for Hofstadter’s MIU-System by Swanson and McEliece in The Mathematical Intelligencer 10 (2), Spring 1988, pp. 48-9.
- 6.
When arithmetic is developed within set theory, all five postulates come into play.
- 7.
There is no unanimity among mathematicians about whether \(\mathbb {N}\) should include 0 or start with 1. Even Peano waffled on this. We’re including 0 for reasons already mentioned. We’ll use \(\mathbb {N}^{\,+}\) to represent the set of positive natural numbers.
- 8.
We’re consciously avoiding the notation \(x + 1\) (or even \(x^+\)) for the successor of x since we’ve yet to define addition, which will be done in terms of the successor function S.
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Jongsma, C. (2019). Mathematical Induction and Arithmetic. In: Introduction to Discrete Mathematics via Logic and Proof. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-25358-5_3
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DOI: https://doi.org/10.1007/978-3-030-25358-5_3
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