How was the axiom of strong mathematical induction misapplied in the proof of the following obviously false statement?
Theorem." All rabbits are the same color.''
Proof. Let P(n) denote the following statement:
Any given n rabbits are the same color.
If we can show that P(n) is true for all n belongs to N, then we will clearly have proven that all rabbits are the same color. We proceed by induction. Clearly, if we have only one rabbit, then that rabbit will be the same color as itself. Assume that for all j k, any given j rabbits are the same color. We must show that any given k + 1 rabbits are the same color. We start by picking an arbitrary collection of k + 1 rabbits. Number the rabbits 1 to k + 1. If we consider the rst and (k + 1)th rabbits, then these two must be the same color by the induction hypothesis. Since the rst rabbit and the second through kth rabbits are all the same color by the same reason, we must conclude that all k + 1 rabbits are the same color. Finally, the axiom of strong induction tells us that any given n rabbits are the same color".
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