The problem with
Sand Jack's `
proof' is that it improperly uses
mathematical induction. We have
- The length of figure 0 (Sand Jack's first figure) is 2.
- If the length of figure k is 2, then the length of figure k+1 is 2.
From this, mathematical induction allows us to establish that the length of figure
n is 2, for any natural number
n. However,
induction does not allow us to make such a claim for the limit. Sand Jack's proof is
parallel to the following, also incorrect, one:
Consider the sequence 1, 1/2, . . ., 1/n, . . . . Clearly, each element of this sequence is greater than zero. Hence the limit of this sequence (that is, zero) is greater than zero. Therefore, zero is greater than itself.
I have heard Sand Jack's proof attacked with ``the limit of the sequence of figures is not a straight line''. However, the limit is a straight line, for a reasonable definition of `limit': for each figure n, let f_n be the natural parametrisation of that figure. Then the sequence { f_n } converges (uniformly) to the natural parametrisation of the diagonal line.
To reiterate: after any finite number of steps, the
length is indeed 2, but the figure is not a straight line. In the limit, the figure does become a
straight line (not a
fractal as
Dhericean supposes), but the length is not 2.
Dhericean and
evan927 state the first part of this, but the
second is just as important.