Infinity is Many-One

Question

Bertrand Russell in Introduction to mathematical philosophy states, "It will be observed that zero and infinity, alone among ratios, are not one-one. Zero is one-many, and infinity is many-one." (P.40)

I think I understand that zero is one-many because I can take zero divided by any non-zero number and the conclusion is the same. I don't understand that when he says "infinity is many-one"

Answer 1

See the full discusssion in CH:VII RATIONAL, REAL, AND COMPLEX NUMBERS; he is defining fractions :

We shall define the fraction $\dfrac m n$ as being that relation which holds between two inductive numbers [see page 27 : We shall use the phrase “inductive numbers” to mean thesame set as we have hitherto spoken of as the “natural numbers.”] $x, y$ when $xn = ym$. This definition enables us to prove that $\dfrac m n$ is a one-one relation, provided neither $m$ nor $n$ is zero.

It will be seen that $\dfrac 0 n$ is always the same relation, whatever inductive number $n$ may be; it is, in short, the relation of $0$ to any other inductive cardinal. [Thus :] Zero is one-many.

Conversely, the relation $\dfrac m 0$ is always the same, whatever inductive number $m$ may be. There is not any inductive cardinal to correspond to $\dfrac m 0$. We may call it “the infinity of rationals.” It is an instance of the sort of infinite that is traditional in mathematics, and that is represented by “$\infty$.” [Thus:] infinity is many-one.

In other words, if we agree to "name" with the symbol “$\infty$” the relation $\dfrac m 0$, this holds for any $m$.

Answer 2

Consider the Real interval $[0,\epsilon]$. How many points are inside that interval? Divide the interval so that the new interval is $[0,\epsilon/2] $. How many points are inside this interval? Still infinitely many. We say that $\mathbb{Q}$ is dense in $\mathbb{R}$ in the language of Real Analysis. This means you can always find rational numbers in a neighbourhood of a real number (loosely speaking). In fact, you can always find infinitely many rational numbers inside an arbitrarily small (non zero) real interval