A faulty induction, the horses

Observation — Typical error: a faulty "proof" of a false statement

There is the famous “proof” that in every group of $n$ horses the horses all have the same colour. The base ($n=1$) is obvious; the inductive step takes a group of $k+1$ horses, removes one to obtain $k$ (same colour), then puts it back and removes another, comparing it with the remaining $k$.

The flaw is that the argument requires the two subgroups of $k$ horses to intersect, which fails for $k=1$: induction requires the chain $P(1)\to P(2)\to P(3)\to\ldots$ to be unbroken, a single arithmetic step is not enough.

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Topics: Set theory
Concepts: Principle of induction
Methods: Induction