Lesson 4

Thus, the real line has cardinality greater than countable (even though we haven't formally defined when one cardinality is greater than another). It is natural to assume that the cardinality of points in the plane will be greater than the cardinality of the line. At the very least, this agrees with our intuitive notions of dimension and volume. It turns out, the plane and the line are actually equinumerous. This surprised even Cantor himself, who proved it.

Theorem. The segment \([0,1]\) is equinumerous with the square \([0,1] \times [0,1]\).

Proof. The square is equinumerous with the set of ordered pairs of infinite binary sequences (by the theorem above). To a pair of sequences \(((a_{1},a_{2}, \dotsc), (b_{1}, b_{2}, \dotsc))\) we associate the sequence \((a_{1},b_{1},a_{2},b_{2},\dotsc)\).◼