Counting is one of the basic elementary mathematical activities. It comes with two complementary aspects: to determine the number of elements of a set - and to create an ordering between the objects of counting just by counting them over. For finite sets of objects these two aspects are realized by the same type of num- bers: the natural numbers. That these complementary aspects of the counting pro- cess may need different kinds of numbers becomes apparent if one extends the process of counting to infinite sets. As general tools to determine numbers of elements the cardinals have been created...

Counting is one of the basic elementary mathematical activities. It comes with two complementary aspects: to determine the number of elements of a set - and to create an ordering between the objects of counting just by counting them over. For finite sets of objects these two aspects are realized by the same type of num- bers: the natural numbers. That these complementary aspects of the counting pro- cess may need different kinds of numbers becomes apparent if one extends the process of counting to infinite sets. As general tools to determine numbers of elements the cardinals have been created...