The Well-Ordering Principle as an Alternative to Mathematical Induction in Our Lower Division Recursive Formula Proofs

Abstract

The principle of mathematical induction is possibly too sophisticated a mathematical topic to teach many two-year college mathematics students with the desired degree of understanding that is essential to good mathematics education. Perhaps we should remove the teaching of mathematical induction from our high school and lower division mathematics curriculum and replace it by using the more basic notion of the Well-Ordering Principle. Mathematical induction can then wait for the upper division course and a more proper mathematical treatment. This could be the rather sophisticated presentation of the Theorem of Induction by using the Well-Ordering Principle. The student would have been sufficiently grounded in the Well-Ordering Principle and its use by this time that induction would naturally follow. The following discussion illustrates how recursive type formulas might be verified in the lower division mathematics courses. The method actually parallels that commonly used with induction. First state the Axiom of the Least Positive Integer: Given any set of positive integers, there is a smallest or least element. After looking at several examples and explaining the meaning of the term axiom, use the axiom in establishing (in an indirect proof, as compared to the direct proof used in induction) several recursive formulas such as 1 + 2 + ... + n = n(n + 1)/2, or possibly yn = P(1 + .06), where yn is the amount of money accumulated in a savings account compounded annually at 6% after n years with no withdrawals, and an initial investment of P dollars. The proof of the first formula is given to illustrate the method employed.