Segment and Square

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Segment and Square

Summary

The guide explains one of Georg Cantor’s most surprising discoveries in set theory: a line segment and a square contain the same number of points, despite the square appearing much larger. Cantor originally tried to prove that the square had a greater infinity of points, but instead found a one-to-one correspondence between the two sets. The construction represents each point in the square by its decimal coordinates ($(x, y)$), then creates a single number on the segment by interleaving the digits of the two decimal expansions. 

This process uniquely maps almost every point in the square to a point on the segment, illustrating that both sets have the same cardinality, known as the cardinality of the continuum. Although the proof requires careful handling of non-unique decimal expansions, the central idea is remarkably elegant and counterintuitive. The result demonstrates that geometric dimension does not determine the size of an infinite set: not only squares, but cubes and any figure containing a line segment, possess exactly as many points as a single line segment. It is a classic example of how mathematical reasoning can overturn everyday intuition about infinity.

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