Mathematical Induction and Arithmetic
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.