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

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