I recently read about the false inductive proof that *all horses are the same colour*. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2).

After reading this, I now realise I have never really understood mathematical induction. It seems obvious for sums, but when presented with this example I was stumped.

I have some questions:

**Question 1. The inductive step**

for *all horses are the same colour* goes roughly as follows:

Assume that there are n horses, numbered 1 to n. By induction horses 1 through (n-1) are the same colour, and similarly horses 2 through n are the same colour. The overlapping sets H_0 = {1,2,...,(n-1)} and H_1 = {2,3,...,n} imply that horse 1 and horse n are the same colour. Thus all horses are the same colour.

I have an issue with one of the statements:

By induction horses 1 through (n-1) are the same colour, and similarly horses 2 through n are the same colour.

I understand why horses 1 through (n-1) are the same colour (by hypothesis), but why can we just say horse n is the same colour? It seems to imply the numbering is arbitrary to me, but then induction is completely invalid unless we can define an order?

**Question 2. The base case**:

If we grant the inductive step reasoning (of which I am unsure whether that is valid), the induction relies on the fact that for some k, two distinct k-1 subsets have shared elements, from which we can conclude they are all the same colour. So the base case should be n=2 (or should it be n=3?).

However the trick used in the inductive step seems to imply the numbering of the horses is arbritary (as I briefly mentioned above), so a subset of size n=2 would seem to require we prove all subsets of size 2 have horses which are the same colour, which makes the induction step redundant?

**General Question 3.**

How do we choose what inductive steps are valid, and what base cases are required? I realise this is poorly phrased, but is there a way in general, or a set of heuristics to enable a valid basis to be chosen?

**A slightly modified version of Mathematical Induction that seems to me more intuitive**

Perhaps my understanding of induction is incorrect. It seems to me mathematical induction is more intuitive when presented as follows:

For each positive integer n > n_0, Let p(n) be a statement.

Step 1. We first show that If p(k) then p(k+1) is true for every positive integer k. To do this we choose an arbitrary k and show, by direct proof usually, this implies (k+1)

Step 2. Choose a base case n_0 on which we induct on to all n > n_0. Here we use the fact that we have shown that for an arbritrary k, k+1 is implied. Choosing a valid basis allows the "induction domino effect" to work its magic.

Then we have shown p(n) is true for every positive integer n > n_0

That seems to me easier to understand. Although do we need to explicitly show our base case implies n_0 + 1?

I suppose this is a different way of asking the same question I asked above, namely how do we choose valid base cases?

Hopefully this question isn't too rambling. Please let me know if anything requires clarification.