What is wrong with the following inductive argument intended to show that:

every set of lines in the plane, no two of which are parallel, meet in a common point?

The argument uses induction on the number n 2 of lines.

Basis (n = 2). Hypothesis. Step.

Any two lines in the plane that are not parallel meet in a common point, by definition of parallelism.

Assume that, for some k 2, a set of k parallel lines, not two of which are parallel, meet in a common point.

Suppose we have a set of k + 1 distinct lines in the plane, no two of which are parallel. Let the lines be ordered arbitrarily. By the inductive hypothesis, the first k lines meet in a common point p1 . Similarly, the last k lines meet in a common point p2. If p1 p2, then the k + 1 lines are all the same since two points determine a line in the plane. Hence, p1 = p2, and the statement is true.