linguistic:paraphrasing
Example of error in the induction step

The induction step must be proved for all values of n. To illustrate this, Joel E. Cohen proposed the following argument, which purports to prove by mathematical induction that all horses are of the same color:

Base case: in a set of only one horse, there is only one color.

Induction step: assume as induction hypothesis that within any set of n {\displaystyle n} horses, there is only one color. Now look at any set of n + 1 {\displaystyle n+1} horses. Number them: 1, 2, 3, …, n, n + 1 {\displaystyle 1,2,3,\dotsc,n,n+1}. Consider the sets { 1, 2, 3, …, n } {\textstyle \left\{1,2,3,\dotsc,n\right\}} and { 2, 3, 4, …, n + 1 } {\textstyle \left\{2,3,4,\dotsc,n+1\right\}}. Each is a set of only n {\displaystyle n} horses, therefore within each there is only one color. But the two sets overlap, so there must be only one color among all n + 1 {\displaystyle n+1} horses.

The base case n = 1 {\displaystyle n=1} is trivial, and the induction step is correct in all cases n > 1 {\displaystyle n>1}. However, the argument used in the induction step is incorrect for n + 1 = 2 {\displaystyle n+1=2}, because the statement that "the two sets overlap" is false for { 1 } {\textstyle \left\{1\right\}} and { 2 } {\textstyle \left\{2\right\}}.