### CANTOR'S THEORY

For finite sets, cardinality of a set = number of elements in the set.

Two infinite sets have the same cardinality if their elements can be placed in 1-to-1 correspondence with each other.

The cardinality of **N** is denoted (aleph-null).

Any set of cardinality is called **countable**.

Is **Z** countable ? Yes. e.g. :

**N** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | .......... |

**Z** | 0 | 1 | -1 | 2 | -2 | 3 | -3 | 4 | -4 | ......... |

.
Surely **Q** isn't countable ? Yes it is !

e.g. imagine the whole set of rationals in an infinite array :

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traverse this array with a series of diagonal **V**'s from the middle of the top row downwards. Ignore fractions not in their lowest terms ( and , n > 1). We get :

.**N** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | .... |

**Q** | 0 | 1 | -1 | 2 | | - | -2 | 3 | | - | -3 | .... |

- and eventually every member of **Q** appears in the list. So **Q** is countable.

Maybe *every* infinite set is countable !

What about **R** ?

**R** is uncountable Proof

Algebraic and Transcendental Numbers

Foundations of Mathematics

History of Mathematics Module

Links to other History of Mathematics sites

[ Comments ] [ Module Leader ]

These pages are maintained by M.I.Woodcock.