What is wrong with the proof of the following (true) theorem? theorem 2. all new

Question

What is wrong with the proof of the following (true) theorem?

theorem 2. all new 1922 ford model T cars had the same exterior color.

proof. The case n = 1 is obvious.

Suppose that in any set of n new Model T's, all had the same exterior color. Consider a set of n+1 new Model T's, lined up from left to right.

We may assume by induction that in the set L of the n Model T's

to the left all had the same exterior color, and similarly that in the set

R of the n Model T's to the right all had the same exterior color. But

then evidently all the n + 1 Model T's had the same exterior color, for

the leftmost and rightmost Model T's had the same exterior color as

all the Model T's in between.

By induction, for every number n, in every set of n new Model T's

all had the same exterior color. Since the set of all new 1922 Model

T's was one such set, the theorem is proved.

(Henry Ford was reputed to have said of the Model T, You can

paint it any color, so long as it's black.")

Explanation / Answer

Induction requires that you look at the first n to prove the truth of n+1. You don't do this when you look at the left and right sets. You cannot be sure that they have the same color as the first one you examined. The first one you examinied must be in the set of n.

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