That all numbers in an n- element set of positive integers are equal. Then p(1)

Question

That all numbers in an n- element set of positive integers are equal. Then p(1) is true. Now assume p(n-1) is true, and let N be the set of the first n integers. Let N' be the set of the first n-1 integers. By p(n-1), all members of N' are equal, and all members of N'' are equal. Thus, the first n-1 elements of N are equal and the last n-1 elements of N are equal, and so all elements of N are equal. Therefore, all positive integers are equal.

I know that the answer is The error is in going from p(1) to p(2) where in a 2 element set, the first n-1 integers

and the last n-1 integers don't overlap. But can someone explain to me why?

Explanation / Answer

The issue lies in the second to last statement. For example, let N = {1,2}. N is a 2 - element set such that N' = {1} and N'' = {2}. All elements of N' are equal and all elements of N'' are equal, but N has two distinct elements: 1 and 2.

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