examday

/TMUA

Consider the following proof that everyone on Earth has the same name.

• I. Let $P(n)$ be the statement that every group of $n$ people has the same name. Then $P(1)$ is true since every person in a group of one has the same name as every person in that group.
• II. Assume that for every group of $k$ people, everyone has the same name as everyone else in that group.
• III. Consider an arbitrary group of $k+1$ people $p_1,\ldots,p_{k+1}$.
• IV. By the assumption, the people $p_1,\ldots,p_k$ have the same name since they form a group of $k$ people.
• V. Similarly, the people $p_2,\ldots,p_{k+1}$ have the same name since they also form a group of $k$ people.
• VI. Hence the people $p_1,\ldots,p_{k+1}$ have the same name. But $p_1,\ldots,p_{k+1}$ can be anyone since they come from an arbitrary group, thus $P(k+1)$ is true.
• VII. Since $P(1)$ is true and $P(k)$ implies $P(k+1)$ for all integers $k\ge1$, every group of $n$ people has the same name.

Where is the first error in the proof?