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Thu 8 Sep, 2005 08:03 pm
Claim: All horses are the same color.
Proof (Mathematical Induction):
Clearly all horses in any set of 1 horse are all the same color. This completes the basis step. Now assume that all horses in any set of n horses are the same color. Consider a set of n + 1 horses, labeled with the integers 1, 2, . . . , n + 1. By the induction hypothesis, horses 1, 2, . . . , n are all the same color, as are horses 2, 3, . . . , n, n + 1. Because these two sets of horses have common members, namely horses 2, 3, 4, . . . , n, all n + 1 horses must be the same color. This completes the induction argument.
What's the flaw here?
You're ignoring horses that exist outside of the set. Just because a set is infinite doesn't mean it contains everything.
That the singleton {1} is of the same color horse and the singleton {2} is of the same color horse does not lead to the claim that {1,2} be of the same color horses.
Quote:Now assume that all horses in any set of n horses are the same color.
That's one heck of an assumption.
While you could fulfill that assumption with an infinite supply of horses, you cannot then turn around and apply it to the finite number of horses that are actually in existence.
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