Mathematicae Varietas


Is the Line Segment  [0,1]  Infinite?

Consider the sequence of functions:

Each defined on [0,1].  The graphics above show functions with   n = 2, n = 10, and n = 100. It is easy to prove that  the sequence converges uniformly to the straight line segment [0,1].  It is also straightforward to demonstrate that  the length of each curve is

At any fixed degree of magnification, there is an n so large that the nth curve appears as a straight line. (Mathematicians routinely dismiss this sort of seeming paradox by simply citing "length" as a feature that is not preserved under uniform convergence.)

Note, that for each n the maximum distance of the curve from the line segment [0,1] is 1/10√n .  So for any positive real number  r  there is an  n  such that 0 < 1/10√n < r .  As we pass from the finite to the infinite, entering the strange world of  *R, the resulting object is an infinite line of no thickness - trapped in the interval [0,1] and having a "cloud" of infinitesimal points cloaking it, above and below.  

A very old example related to the one given above, one that confounded mathematicians at the beginning of the 20th century - until they decided to ignore it - is the Diagonal Paradox :

Diagonal Paradox

The large square is one unit on a side. The "stairsteps" going from the bottom left corner to the top right corner contain n=5, n=20, and n=110 steps, respectively. The length of each such staircase curve is exactly 2 units. But as n becomes infinite, the polygonal curves approach a straight line: the diagonal of the large square, which is the square root of 2 (approximately 1.414) units long.  Again, as we pass from the finite to the infinite, the result is a diagonal line of no thickness, 2 units long, trapped within an interval 1.414 units long, and sheathed in a halo of infinitesimal points.

Vardi Ilan (Paris) has an interesting comment about this kind of paradox:
"Regarding the notion of finite curves of infinite length, I have
studied the works of Archimedes and I am fairly convinced that he
had some notion of this possibility, because in his works he is very
careful to set up axioms for length, in particular, his axiom that
if two convex curves have the same endpoints and one is inside the
other, then the inside one will be shorter. Otherwise, you do get

Continue to Page 5