Unendlichkeit - Mathematics - Set Theory - Cardinality of The Continuum

Cardinality of The Continuum

One of Cantor's most important results was that the cardinality of the continuum is greater than that of the natural numbers ; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that (see Cantor's diagonal argument or Cantor's first uncountability proof).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, (see Beth one). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points in the side of a square and those in the square.