Cardinality and Ordinality: life was so easy until zero came along

When we are dealing with the counting numbers — 1, 2, 3, and so on — it is easy to rank them in order.   One is the first counting number, two is the second counting number, and three is the third.  We don’t have to worry about mixing up the value of a number — its cardinality — with the order in which it arrives — its ordinality — since they are essentially the same thing.  For years, this was the state of affairs, and everybody was happy.  But as zero came into the fold, the neat relationship between a number’s cardinality and its ordinality was ruined.  The numbers went 0, 1, 2, 3: zero came first, one was second in line, and two was in third place (i.e. 0 has a cardinal value of 0 and an ordinal value of 1; 1 has a cardinal value of 1 and an ordinal value of 2; 2 has a cardinal value of 2 and an ordinal value of 3).  No longer were cardinality and ordinality interchangeable.

Zero, the Biography of a Dangerous Idea by Charles Seife

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