# Is zero an ordinal or cardinal number ?

Peter Harremoës wondered whether zero is a natural number, arXiv:1102.0418v1 [math.HO]. He starts out by admitting that it is a matter of definition, but then proceeds with the issue of ordinal and cardinal numbers, so perhaps I should rephrase his true question as I did now in my own title above or to the left. A google on “Is zero a natural number ?” and “Is 0 a natural number ?” generates some 15,000 hits. A bit to my amazement there are more people pondering the question (though close to 0.0% of mankind in statistical approximation (in writing and not reading)).

Harremoës approaches the issue from a set-theoretic point of view, though visits Kindergarten with some nice observations. My focus is didactics, and thus I will begin with Kindergarten.

Kids learn the sequence S = {1, 2, 3, … }, and tally off their fingers. When they count the elements of a set A (apples), they bring the elements in A in a one-to-one relation with the elements in S, and also in that order {1, 2, 3, … }. The last element counted gives the total number of elements in A.

For terminology, an ordered sequence gives an ordinal measure, while the total gives the cardinal measure for the number of elements in a set.

The kids learn also the sequence O = {1st, 2nd, 3rd, ….}. Now this is interesting ! It is rather this list O that gives the ordinals ! Rather than saying “This is apple one, this is apple two, this is ….” they learn to say “This is the first apple, this is the second apple, this is …”.

When counting elements in a set A then it does not matter in which order the elements are put – and the cardinal number has the property that it has the same value in whatever order its elements are put. But, for O, the order must follow the order of the elements of the set A that is being considered.

Thus, kids first learn ordinals and cardinals in a mixed manner with S. Then the ordinals are created separately in O. Then S becomes important for the cardinals. The introduction of O requires a bit of unlearning alongside learning.

 Set A is counted using ordered {1, 2, 3, …} Order in A is not relevant Order in A is relevant Counting (process) (“Order some or all.”) {1, 2, 3, ….} for first training {1st, 2nd, 3rd …} Cardinal (result) (“How many elements are there ?) {1, 2, 3, ….} {1, 2, 3, ….}

With this established, I think that I must object to the use of the idea that “ordinal numbers” and “cardinal numbers” would be separate sorts of numbers. We only have the numbers S. The “cardinal number of a set A” is just idiom to identify the number of elements, but this does not suggest that “cardinal number” is a specific kind of number (like rational number or complex  number).  It is less common to speak about the “ordinal number of an element”. While “What number are you in line ?” indicates such ordering, still you are not a number, and “ordinal number” is not a special number but merely S applied to ordering.

Thus “Is zero an ordinal or cardinal number ?” is a nonsensical question. Zero can be the value of the cardinal number of a set. Whether you start counting with 0 is an issue of convenience, and probably not practical in Kindergarten (but this needs testing). The true issue at hand is not quite arithmetic but actually part of the theory of measurement, with nominal, ordinal, interval and ratio scales.

When you have a list of elements, it is not so practical to start the labeling with 0, since the rank numbers might become adjectives that differ from the proper ranks. For example, Pierre van Hiele, in his masterly exposition on didactics, labeled his levels of understanding by starting at a base or zero level, counting on to level 4. In terms of rank, the first level would be level 0. However, the tendency would be to associated “level 3” with “the third level”, with “third” the adjective of “three”. It appears difficult to suppress that tendency. Hence it is better to start lists with label 1. (In inverted manner, the calendar has no year 0 but it has a first year.)

Note that Dutch has different words for number (“getal”, as in the list of natural numbers, or the pure decimal system, old-English “tale”) and cardinal number (“aantal”, the number of elements, English “tally”). Teaching in Dutch is a bit easier than in English. See Google Translate on this. In etymology we can find a curious connection of counting with  speech itself. To give an account or reckoning may consist of a story or a list of numbers: “Origin of tale: (Webster) Middle English ; from Old English talu, speech, number, akin to German zahl, number, Dutch taal, speech ; from Indo-European base an unverified form del-, to aim, reckon, trick from source Classical Greek dolos, Classical Latin dolus, guile, artifice”. In Dutch the subtle distinction is between “tellen” (to count) and “vertellen” (to tell).

Let us consider kids in Syria, Israel, Gaza or Ukraine who have N[10] = {0, 1, 2, 3, …, 10} fingers left. Since the places of these fingers on their hands or the way how these fingers are ordered doesn’t matter, the natural point of view is that the numbers are cardinal values (“how many left ?”). There also arises a natural order that one set is larger than another, and with thus cardinal values 0 < 1 < 2 < 3 … < 10.

The Egyptians already had a symbol for “none”. They didn’t regard it as a number though. However once you set up a system of arithmetic then it becomes convenient to regard 0 as a number, so that 1 – 1 = 0. Slowly language adapts to that use. Given the naturalness of the question for kids to ask “How many fingers do I have left ?”, thus with the focus on cardinality, and given that the answer may also be “zero”, it is more reasonable to include 0 in the list of numbers that are regarded as natural, giving = {0, 1, 2, … }.

However, when you know that a set is non-empty, then you can use N+ = S for the positive integers. Hopefully the little ones still have a thumb to suck on, to fall asleep, and be ready for another day of counting.