The Universal Computer

Cantor’s Quest for Infinite Numbers

then

A = B =4 and C =2.

Of course it is easy to set up a one-to-one matching between A and B:

♣♦♥♠



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What happens when two sets do not have the same cardinal number?

In symbols, it is a question of sets M and N such that

M = N.Insuch

a case, one of the two cardinal numbers is the larger and the other the

smaller. Using the standard symbols < (“is less than”), and > (“is greater

than”) we can write

M<N (or equivalently, N>M) to indicate that it

is N that has the larger cardinal number. To prove that this is indeed the

case what is needed is a one-to-one match-up between M and some subset

of N.

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Thus, in the example ...