# Do we count from 0 or from 1?

### The ordinal use of cardinal expressions

ordinal number

noun
Date: 1607

1. a number designating the place (as first, second, or third) occupied by an item in an ordered sequence
2. a number assigned to an ordered set that designates both the order of its elements and its cardinal number
cardinal number

noun
Date: 1591

1. a number (as 1, 5, 15) that is used in simple counting and that indicates how many elements there are in an assemblage
2. the property that a mathematical set has in common with all sets that can be put in one-to-one correspondence with it
Note: the definitions 2 of ordinal and cardinal numbers stem from the development of set theory in the 19th century. The main motivation of these definitions is in the definition of infinite numbers. In the remainder of this note I use definitions 1 only.

There is a technical difference between cardinal and ordinal numbers. The distinction can be seen in the way that these numbers are used. A number used to designate the size of a set--i.e., to answer the question, "How many?"--is used cardinally. Any use that depends on the position of the number in the prescribed sequence is the ordinal use of the number. The number found at the top or bottom of a page in a book is an example of the ordinal use of the number. ["set theory", Encyclopędia Britannica Online]

A word like "first", "second" or "third" has ordinal use only. Let me call such a word an "ordinal expression". A word like "one", "two" or "three" is mostly used as cardinal number, but in contemporary speech also as ordinal. Examples are "act 1", "take 2", "page 5" and "definition 2". Let us call such words "cardinal expressions". When using cardinal expressions as ordinals, there are two ways to count: starting from 0, or starting from 1. If a play consists of 3 acts, these acts can be called, "act 1", "act 2" and "act 3", or "act 0", "act 1" and "act 2".

In fact, a phrase like "act 2" should be interpreted as a name for the intended act. As people can name things anyway they like, one might also call the acts "Alice", "Bob" and "Casey", or "act 4", "act -1" and "act 666". However, when the objects to be named come in an obvious serial order, naming them 1-2-3 or 0-1-2 appears most sensible.

If I propose to a date to meet me on "bench 2 past the purple fountain" I'm literally speaking dependent on an authority that has called the intended bench "bench 2". In the absence of such an authority it makes much more sense to speak of "the second bench". As in most cases there is no established authority, ordinal expressions are much more often used to denote ordinals than are cardinal expressions.

Take a soccer match. Reporters have a tendency to refer to events that happened in the first minute, the 37th minute, or the 90th minute. This terminology is unambiguous. If a reporter would speak of "minute 89", she might have to explain which minute it is exactly that she chooses to denote by the name "minute 89" (the 89th minute perhaps? Or the 90th?). This issue can entirely be avoided by using ordinal expressions.

Nevertheless, cardinal expressions have their ordinal use as well. If a scientific paper contains many definitions, it is convenient to name and label every one of them (e.g. "Definition 12"), so that one can conveniently refer back.

### Inclusive versus exclusive counting

I suggest to call the naming scheme 1-2-3 "inclusive", and the naming scheme 0-1-2 "exclusive". For suppose that I count 10 apples. When counting the apples inclusively, I point to the first apple and say "one", then I point to the second apple and say "two", etc. If, by doing so, I name the second apple "apple 2", this name indicates that so far 2 apples have been counted, including the one being counted at that very moment. Counting the apples exclusively consists of pointing to the first apple and saying "zero", pointing to the second apple and saying "one", etc. If, by doing so, I name the second apple "apple 1", this name indicates that so far 1 apple has been counted, excluding the one being counted at that very moment.

Note that for ordinal expressions there is no choice in using them inclusively or exclusively. The meaning of such expressions is incorporated in the English language (and in all other languages I know of); they are always used inclusively. There can be no such thing as a 0th apple. If fact, in the previous paragraph I explained exclusive counting by saying `I name the second apple "apple 1"', thereby employing the word "second" inclusively.

Now, suppose that one is in charge of numbering something. Is inclusive or exclusive numbering preferable? My answer is that in most cases it doesn't matter, but in the cases in which it does, inclusive counting is preferable.

As an example, suppose you become pope, and chose the name "John". However, one and only one pope before you had the name John already, and you need to distinguish yourself from him. Do you call yourself "John 1" (thereby implying that pope John was in fact John 0), or "John 2" (implying that pope John was in fact John 1)?

In my opinion, the best choice here is to call yourself "John 2". The main reason is that you will be referred to as "the second pope John", or "John the second" for short, regardless of your choice. Naming the second John "John 1" would be confusing. In short, since the ordinal expressions have fixed meanings already, it is best when choosing cardinal expressions for numbering to let them correspond with the ordinal expressions. This entails inclusivity.

Another example: suppose you are counting an unknown number of apples (neatly arranged in a single-file row). If you use inclusive counting, and when pointing to the last apple you say "10", you know that there are 10 apples, so no concluding statement is needed. Also, when counting the seventh apple, you know that there are at least 7 apples, so calling this apple "7" best describes the information gathered so far.

A third reason to use inclusive counting, is to stick with tradition. Except for recent applications in mathematics and computer science, virtually all of humanity uses inclusive counting. Suppose that somebody started counting a large amount of apples, and at the end of his shift, his colleague entered to relieve him, and heard him say "125", while throwing a particular apple in the basket. Then the colleague continues by numbering the next apple "126", and the process ends at "188". At that point the colleague has to guess whether the counting started at 0 or at 1, i.e. whether the total number of apples is 188 or 189. In the absence of further information her best guess would be that the counting started at 1, since almost nobody starts counting at 0.

Thus, if for any reason a default numbering system has to be chosen, inclusive counting (i.e. starting at 1) appears to be preferable over exclusive counting.

### What are the consequence of this for the millennium problem?

First of all, if all we would know about the current year is that it is called "year 1999", it makes more sense to postulate that it is the 1999th year than that it is the 2000th year.

Secondly, in the first centuries of the current calendar, cardinal numbers were not used ordinally at all. For ordinal purposes people used ordinal numbers. Thus rather than speaking about the year 1, the year 527, or the year 1999, people talked about the first year, the 527th year, or the 1999th year. By doing so, they avoided all ambiguity about the point where the counting starts. As we are now in the 1999th year (that only recently has been called "the year 1999"), the start of the 3rd millennium is still more than a year away.

 Rob van Glabbeek December 6, 1999 rvg@cs.stanford.edu