Interesting examples of vacuous / void entities

Question

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty set is a non-empty set):

"Perhaps as a result of studying set theory, I was surprised when I learned that some respectable combinatorialists consider such things as this to be mere convention. One of them even said a case could be made for setting the number of partitions to 0 when $n=0$. By stark contrast, Gian-Carlo Rota wrote in [Rota2], p.~15, that 'the kind of mathematical reasoning that physicists find unbearably pedantic' leads not only to the conclusion that the elementary symmetric function in no variables is 1, but straight from there to the theory of the Euler characteristic, so that 'such reasoning does pay off.' The only other really sexy example I know is from applied statistics: the non-central chi-square distribution with zero degrees of freedom, unlike its 'central' counterpart, is non-trivial."

The cited paper was: G-C. Rota, Geometric Probability, Mathematical Intelligencer, 20 (4), 1998, pp. 11–16. The paper in which my footnote appears is the first one you see here, doi: 10.37236/1027.

Question: What other really gaudy examples are there?

Some remarks:

• From one point of view, the whole concept of vacuous truth is silly. It is a counterintuitive but true proposition that Minneapolis is at a higher latitude than Toronto. "Ex falso quodlibet" (or whatever the Latin phrase is) and so if you believe Toronto is a more northerly locale than Minneapolis, it will lead you into all sorts of mistakes like $2 + 2 = 5$, etc. But that is nonsense.

• From another point of view, in its proper mathematical context, it makes perfect sense.

• People use examples like propositions about all volcanoes made of pure gold, etc. That's bad pedagogy and bad in other ways. What if I ask whether all cell phones in the classroom have been shut off? If there are no cell phones in the room (that is more realistic than volcanoes made of gold, isn't it??) then the correct answer is "yes". That's a good example, showing, if only in a small way, the utility of the concept when used properly.

• I don't think it's mere convention that the number of partitions of the empty set is 1; it follows logically from some basic things in logic. Those don't make sense in some contexts (see "Minneapolis", "Toronto", etc., above) but in fact the only truth value that can be assigned to "$F\Longrightarrow F$" or "$F\Longrightarrow T$" that makes it possible to fill in the truth table without knowing the content of the false proposition (and satisfies the other desiderata?) is $T$. That's a fact whose truth doesn't depend on conventions.

Answer 1

Maybe it is not serious, but then in a sense none of the examples in this thread are.

Jordan decomposition represents every operator on a finite-dimensional space in a unique way as $S+N$ where $S$ is a diagonalizable operator, $N$ is a nilpotent operator, and $SN=NS$.

The Jordan decomposition of the zero operator is $0+0$. It thus is the only operator which is diagonalizable and nilpotent at the same time.

Similarly, the zero Lie algebra is the only one which is both semisimple and nilpotent. And one might also argue that it is not simple. Or is it?...

Answer 2

A The empty set is a covering map of any topological space. More generally, a covering map needn't be surjective (although many books claim just that). For example the inclusion of a closed and open subset of a space is a covering. Strangely, I would argue that this entails that the empty topological space, although connected, is not simply connected.

B Dually, given a field $K$, the zero algebra over $K$ is diagonal and in particular étale: the morphism of affine schemes $\varnothing \to \operatorname{Spec}(K) $ is étale. In the same vein, a nonzero constant polynomial over $K$ is separable (its nonexistent roots in an algebraic closure of $K$ are certainly distinct) . We may then say without any exception that the $K$-algebra $K[X]/(f(X))$ is étale iff $f(X)$ is a separable polynomial.

Answer 3

I suppose another example would be those proofs by induction in which you don't need a basis, although no nice examples come to mind instantly. Here I mean proofs in which you show that if $P(m)$ for all $m < n$, then $P(n)$. You don't need to show $P(n)$ holds for the smallest value of $n$, since it is vacuously true that $P(m)$ holds for all smaller values, and therefore the thing proved in the inductive step entails the smallest instance as a special case.

This works not only for natural numbers, but for infinite ordinals.

Answer 4

If we need to define the value of the Euler's function $\varphi$ at infinity, the best choice will be $\varphi(\infty)=2$ This is because $\varphi(n)$ is the number of generating elements in the cyclic group of order $n$, and so if $n\to \infty$, the the cyclic group tends to $\mathbb{Z}$ which has just two generating elements.

$0^0=1$ is another example, which is easy to prove for the natural zero, but it is not true for the real zero!

Answer 5

There are a number of examples from logic.

Many-sorted first-order logic with 0 sorts is propositional logic. A 0-ary function symbol is a constant symbol, and a 0-ary relation symbol is a proposition. These “degenerate” cases are actually quite interesting and important.

Answer 6

Is the span of the empty set in a vector space equal to $\lbrace 0\rbrace$, or does it have no span? The "correct" answer in my opinion is the latter. See Example 3.10.3 of https://web.archive.org/web/20201101123724id_/http://math.mit.edu/~rstan/ec/ec1.pdf for a reason:

3.10.3 Example. We give one simple example of the use of equation (3.34). We wish to count the number of spanning subsets of $V_n$. For the purpose of this example, we say that the empty set $\emptyset$ spans no space, while the subset $\{0\}$ spans the zero-dimensional subspace $\{0\}$.* If $W \in B_n(q)$, then let $f(W)$ be the number of subsets of $V_n$ whose span is $W$, and let $g(W)$ be the number whose span is contained in $W$. Hence $g(W) = 2^{q^{\dim W}} − 1$, since $\emptyset$ has no span. Clearly $$ g(W) = \sum_{T \le W} f(T) $$ so by Möbius inversion in $B_n(q)$, $$ f(W) = \sum_{T \le W} g(T) \mu(T, W) $$ Putting $W = V_n$, there follows $$ \begin {aligned} f (V_n) &= \sum_{T \in B_n(q)} g(T) \mu(T, V_n) \\ &= \sum_{k=0}^n {\boldsymbol n \choose \boldsymbol k} (-1)^{n-k} q^{n - k \choose 2} ( 2^{q^k} - 1 ). \end {aligned} $$

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* The standard convention is that the empty set spans $\{0\}$. If we wish to retain this convention, then we need to enlarge $B_n(q)$ by adding $\emptyset$ below $\{0\}$.

On the other hand, a reason (which I find unconvincing) for the span to be $\lbrace 0\rbrace$ is given by PBRMEASAP at https://www.physicsforums.com/archive/index.php/t-84017.html. This site has a discussion of whether the empty set is a vector space. The correct answer is that it isn't, because one of the axioms is the existence of an additive identity 0.

Update. I agree with the comments that the span of the empty set is $\lbrace 0\rbrace$. What I said above was foolish.