The inductive hypothesis P(n): For any counting number n in N, and set of billiard balls with n members, all the balls have the same color.
Consider any set A of n+1 balls, and the subsets
B=(first n balls), C=(last n balls)
The inductive hypothesis applies to both B and C, so all balls in B have the same color, and likewise for the balls in C. Since the 2 sets have a ball in common, all the balls in their union A=BUC have the same color, proving that P(n+1) is true. By the Induction Axiom, P(n) is true for all counting numbers n so all billiard balls have the same color.
The conclusion is absurd. Can you spot the "error" in this proof?
The Attempt at a Solution
The end says that ALL billiard balls have the same color. But this is true only for the union, not the intersection, of B and C, so not ALL balls apply.
You can have A'=B'UC'=empty set and still fulfill the requirement that all of B' is one color, and all of C' is one color, yet they are not the same color as each other.
Is my "error" finding correct?