Undefined vs Indeterminate in Mathematics

In mathematics, term undefined may characterize an object in two circumstances: a pedestrian situation in which an object has not been defined - perhaps, as yet; and a situation in which an object can't in principle be defined meaningfully. Impossibility of "division by zero" is a manifest example of the latter case. For, say, a real or complex $a\ne 0,$ there is no fruitful way to define $a/0.$ In most number and algebraic systems, zero does not and cannot have a multiplicative inverse, such that $a/0$ is left undefined. (Occasionally, when there is a single "infinite" element $\infty$ as it happens in real or complex projective geometry, one defines $a/0=\infty.)$

For $a=0,$ $a/0$ falls into a different category; $0/0$ is referred to as indeterminate, which means that, depending on the circumstances, the expression $0/0$ may be defined, or may be left undefined as a matter of experience.

An aside: quite often the limit $\displaystyle\lim_{x\rightarrow 0}\frac{f(x)}{g(x)},$ where $\displaystyle\lim_{x\rightarrow 0}f(x)=\lim_{x\rightarrow 0}g(x)=0$ appears in this context. I find this confusing and only accept $0/0$ as a tag that helps distinguish this case from $\displaystyle\lim_{x\rightarrow 0}\frac{f(x)}{g(x)},$ where $\displaystyle\lim_{x\rightarrow 0}f(x)=\lim_{x\rightarrow 0}g(x)=\infty$ which is then described as $\infty /\infty.$ Limits of functions are in the purview of calculus, where limits like the above are calculated by special techniques, e.g., factoring or L'Hôlpital's rule, never by direct substitution.

A classical case of an indeterminate expression is "zero to the power of zero," $0^0.$ If looked at as a limit value $\displaystyle\lim_{x\rightarrow 0}x^{0}$ it ought to be declared $1.$ On the other hand, as the limit value $\displaystyle\lim_{x\rightarrow 0}0^{x}$ $0$ appears more suitable. Function of two variables $f(x,y)=x^y$ is identically $1$ on the positive $x$-axis and is identically $0$ on the positive $y$-axis. So, in calculus, the expression $0^0$ is mostly left undefined (but may be used to tag the class of limits $\displaystyle\lim_{x\rightarrow 0}f(x)^{g(x)},$ where $\displaystyle\lim_{x\rightarrow 0}f(x)=\lim_{x\rightarrow 0}g(x)=0.$

In combinatorics, on the other hand, it was found convenient to set $0^{0}=1$ because, for example, this allows expressions like $\displaystyle (x+y)^{n}=\sum_{k=0}^{n}x^{k}y^{n-k}{n \choose k}$ without making exceptions for $k=0$ or $k=n.$ Also, $m^n$ counts the number of ways to map a set of $n$ elements into a set of $m$ elements. If $n=0,$ so that the first set is empty, there is only one way to map it anywhere, even to "another" empty set. (As a reasoning goes, there is only one way to do nothing.) Thus, it is commonly accepted that "an empty product" is equal to $1.$

By analogy, an empty sum should be defined as $0$ but, in principle, is an indeterminate object. If so, what could be said about the average of the elements of an empty set: is it undefined or indeterminate?

Grégoire Nicollier (University of Applied Sciences of Western Switzerland) has kindly communicated the following observation:

On page 147 of Flughan och evigheten (The Fly and Eternity, published in 1999, no English translation), the Swedish author Håkan Nesser (well-known for his crime novels about Inspector Van Veeteren) describes the opening of the academic year at the local university:

The first days of autumn. Summer sultriness merging into days of full clarity and of vast expanse. The noisy return of the students. The population of the city has suddenly doubled and the average age has been halved.

Question to an attentive reader: How old are the students?!

You may want to figure out by yourself what pulled Grégoire's attention. To be on sure side I'll refresh below a few properties of the average.

Given a set $X=\{x_{1},x_{2},\ldots,x_{n}\}$ of $n$ elements (a better term would be multiset because $X$ may have repeated terms), the average of the set $X$ is defined as $M(X)=\displaystyle\frac{1}{n}\sum_{i=1}^{n}x_{i}.$

Let $Y=\{y_{1},y_{2},\ldots,y_{m}\}$ be a set with $m$-elements. Then, taking the union of two multisets,

$M(X\cup Y)=\displaystyle\frac{nM(X)+mM(Y)}{n+m}.$

Indeed, $nM(X)=\displaystyle\sum_{i=1}^{n}x_{i}$ and $nM(Y)=\displaystyle\sum_{i=1}^{m}y_{i},$ implying that $nM(X)+mM(Y)$ is the sum of all the elements in $X\cup Y.$ The number of such elements is $n+m,$ and the formula follows.

If $n=m$ we obtain an apparent simplification:

$M(X\cup Y)=\displaystyle\frac{nM(X)+nM(Y)}{n+n}=\frac{M(X)+M(Y)}{2}.$

The notable point here is that, for $X$ and $Y$ of equal size,


$M(X\cup Y)=\displaystyle\frac{M(X)+M(Y)}{2},$

independent of actual size of the two sets.

Another worthy property of the average is that, if every element of a set has shifted by the same amount, the average is shifted by the same amount:



Now, I'll proceed with Grégoire Nicollier's observation. As never actually happens in real life or meant in statistical reports, when a population is said to have doubled, there bound to be a margin of error, which suggests that the quote should not be taken literally, for the face value. No doubt, however, Grégoire's intention was for an exact, abstract identity.

Let $P=\{p_{1},p_{2},\ldots,p_{n}\}$ be the set of ages of all the inhabitants of the city in question. Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be the set of all ages of the student population at the local university. The quote from Håkan Nesser together with (1) show that

$M(P\cup S)=\displaystyle\frac{M(P)+M(S)}{2}=\frac{1}{2}M(P),$

from which $M(S)=0.$ With a further assumption that we deal with ages expressed with whole numbers, $M(S)=0$ means that the set of students is empty, its size $n=0$ and, so, $P$ is also empty - the city is a ghost city with no inhabitants, and the local university is an abandoned institution, with no students, let alone tenured staff.

So the two sets $P$ and $S$ are empty, their size is $0,$ and the sum of the ages of their elements is also $0.$ We also found that $M(P)=M(S)=0.$ Thus, arguably,

$0=M(P)=\displaystyle\frac{\displaystyle\sum_{p\in P}p}{0}=\frac{0}{0}.$

Which appears to assert that the average of the elements of an empty set is $0.$ Grégoire Nicollier came up with an objections based on the identity (2). Since all empty sets are he same, $\emptyset +a=\emptyset,$ which, if the average of its elements were to be defined, would imply $M(\emptyset)=M(\emptyset+a)=M(\emptyset)+a,$ for any number $a,$ which is absurd, with the conclusion that the average of the empty set should be declared undefined not just indeterminate. I tend to agree, especially because, I see no useful purpose in declaring otherwise. Still, I am amazed that under the assumptions (concerning the ages and accuracy of population estimates), the quote has more than just one unintended consequences.

So, is the derivation above of $\displaystyle 0=\frac{0}{0}$ meaningful or not. It is questionable in the very least because (1) has been obtained under an implicit assumption that $n\ne 0.$ Employing (1) with the empty sets is akin to getting $1=2$ from $1-1=2-2$ which is a known fallacy, usually better disguised

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