Induction

Induction, a.k.a. Mathematical Induction, is a technique in proofs that lets us prove something for all the integers at and above a starting point.

You must prove the something for the base case. The base case is the lowest value integer that is of interest to you.

You must prove that if this something is true for integer n, then it is also true for the next integer, n+1.

Assume we started with 1. We do the work to prove something for 1, so we have 1. Next, we do the work to prove that having integer n gives us the integer n+1. Now, let’s look at what “n gives n+1” provides for us:

• if n=1 then n+1=2, so having 1 gives us 2
• if n=2 then n+1=3, so having 2 gives us 3
• if n=3 then n+1=4, so having 3 gives us 4

This process gives us all the integers higher than 1 and we might compare it to knocking down an infinite line of dominoes.

I’ll give a small example. Say we are handed two rules:

• One is positive because it is an integer greater than zero.
• An integer greater than a positive integer is also a positive integer.

The first rule gives us “yes for 1”.

Because of Induction, the second rule gives us 2, 3, 4, …

Appendix A

There might be some resistance in your mind, it’s fairly obvious that we concocted the two rules given us for the express purpose of being able to do what we did, and yes, in a way, that can be viewed as cheated.

The deal is, mathematicians had to do precisely that–they had to make choices as to what things would be used as axioms. Most of the time, as a student you will be asked to adhere to whatever they decided (I’m guessing quite a bit of this work was done between 1800 and 1900). However, here at Mockingbird Academy we WILL give you time to think up your own system, write your own rules, and hopefully have some fun doing so.

[As part of the fiction story, the two students had fun discovering that we could use axioms that we chose to prove the three things that are usually called the Axioms of Equality.]