Zero and infinity

Question

Introduction [can be skipped without loss of generality]. This question was closed and, recently, deleted, perhaps for good reason. It did have an answer with 10 upvotes, and another (mine) with 15 upvotes. So that my answer not be lost to m.se, I pose here a version of the deleted question, and my answer.

Question: Is there any situation in Mathematics where there is a sense in which zero and infinity are identified?

Note: although I am posting my own answer, I would encourage others to post theirs as well.

Answer 1

In the lattice of divisibility, where we preorder integers by saying that $a \preceq b$ means that $b$ is divisible by $a$, the integer $0$ is the largest element: e.g. we have

$$ 1 \prec 2 \prec 6 \prec 24 \prec 120 \prec 720 \prec 5040 \prec \ldots \prec 0 $$

(I've chosen the factorials as the example sequence, because every nonzero integer divides some factorial)

There are a few other closely related situations where $0$ fits in the role where one would expect something infinite.

(a preorder is a reflexive, transitive relation. Put differently, a preorder is a partial order where we allow distinct elements to compare equally; e.g. $-2 \preceq 2 \preceq -2$. A partial order is like a normal ordering, except that we don't require every pair of numbers to be comparable. e.g. $2 \not\preceq 3$ and $3 \not\preceq 2$)

Answer 2

Topologically speaking, we can take the real line and give it two endpoints $+\infty$ and $-\infty$. This is called the extended real line.

Then, we can simply decree that $0$ is identified with $+\infty$ and with $-\infty$. The resulting topological space is simply a figure $8$. Or, for extra amusement, you might say it is $\infty$-shaped.

Answer 3

There are some situations in higher mathematics where zero and infinity are identified, sort of.

1. The points on the graph of $y^2={\rm\ a\ cubic\ in\ }x$ form an abelian group when addition is defined by saying that three points add to zero if they are collinear. The zero element of this group is the point at infinity. The keyphrase for learning more about this is "elliptic curve".

2. In the upper-half-plane model of hyperbolic geometry, the "points at infinity" are the points on the $x$-axis, that is, the points with $y$-coordinate zero.

In both cases, it would be misleading to say, "zero equals infinity". In the first case, the point that acts the way you would expect zero to act is physically located at infinity; in the second case, the points that should be infinitely far away have been brought down to height zero.

Answer 4

There are a few occasions where $0$ and $\infty$ would play more or less the same role, depending on the point of view. For instance the characteristic of a (unitary) ring $R$ is defined as the non-negative generator of the kernel of the unique morphism $f:\Bbb Z\to R$, and if that morphism is in fact injective then one therefore says the characteristic of $R$ is $0$ (being the only element of the kernel, there is no other generator than$~0$ to choose from; it could however be argued that the zero ideal does not need any generators at all, just like a $0$-dimensional subspace of a vector space has an empty basis). But one could alternatively define the characteristic as the order of $f(1)=1\in R$ in the additive group of $R$, that is as $\inf\,\{\,n>0\mid f(n)=0\in R\,\}$, which seems a rather natural idea. This definition would have made the characteristic $\infty$ in the case where $f$ is injective.

Answer 5

In the sense of infinite products, a limit of $0$ and a limit of $\infty$ are both considered divergent. That is, $$ \prod_{n=2}^\infty\left(1-\frac1n\right)=0 $$ is considered divergent just as is $$ \prod_{n=2}^\infty\left(1+\frac1n\right)=\infty $$