Proportion between countable infinities?

Question

Suppose we have the set of integers $\mathbb{Z}$ and a congruence class of $x$ modulo $n$, denoted as $[x]_n$ and defined as $\{\ldots,x-2n,x-n,x,x+n,x+2n,\ldots\}$ where $n$ is an integer and $x$ is some real.

The size of both $\mathbb{Z}$ and $[x]_n$ is infinite, but a countably one. If $n>1$ clearly $\mathbb{Z}$ contains more elements than $[x]_n$. More precisely, for each element in $[x]_n$ there are $n$ elements in $\mathbb{Z}$.

Would it be valid to say that $\dfrac{|\mathbb{Z}|}{|[x]_n|}=n$? If $[x]_n\subseteq \mathbb{Z}$ we could say $[x]_n\subset \mathbb{Z}$ because $n>1$. This could be a valid test to test if some set is a proper subset of another one, right?

Answer 1

In set theory we are not allowed to divide set or equivalently division of ordinal numbers. In fact it leads to an ambiguity. We know that $|\mathbb{Z}|=\aleph_0=|[x]_n|=|[x]_m| $ for any $n\ne m$. Your argument $\dfrac{|\mathbb{Z}|}{|[x]_n|}=n $, implies $n=m$ for all $m,n\in\mathbb{Z}$.

In the categories like groups, rings, etc. where a kind of division of sets (quotient groups, quotient ring, etc.) qre defined, you are right. But not in category of sets.

Answer 2

More precisely, for each element in $[x]_n$ there are $n$ elements in $\mathbb{Z}$.

This is only necessarily true if you're picky about how you pair up the elements. It is also possible, for example, to pair the elements up such that for each element in $[x]_n$ there is exactly one element of $\mathbb Z$, namely by $$ f(a) = \frac{a-x}{n} $$ which is a bijection from $[x]_n$ to $\mathbb Z$.

Or you can show that there is exactly one element outside $[0]_n$ for each element in it: $$ g(a) = \begin{cases} a + \lfloor a/n \rfloor & \text{when }a>0 \\ -\bigl(1-a + \lfloor (1-a)/n\rfloor\bigr) & \text{when }a\le 0 \end{cases} $$ which is a bijection from $[0]_n$ to $\mathbb Z\setminus[0]_n$.

So saying that the size of one of the sets is a particular multiple of the other depends on restricting the kind of bijections you're willing to look at. For any particular purpose you may have, this could well be the right thing to do -- but then you're not just looking at the sets as sets, but as sets plus a particular structure on them that allow you to distinguish between "good" and "bad" bijections, and then the usual concept of cardinality is not a useful way to formalize "size" of a set anymore.