TITLE: Is it proper to say that two infinite sets are the "same size" if there is a bijection between them?
QUESTION [5 upvotes]: I get the fact that a set can be called countably infinite if it can be bijected with $\mathbb{N}$, but it feels wrong on many levels to say that they are the same size.
Example: $A=\{x \in \mathbb{N} \mid \exists k\in\mathbb{N}\, (x=2k)\}$ is intuitively smaller than $\mathbb{N}$ because it is $50\%$ in size of $\mathbb{N}$. But it is countably infinite.

REPLY [2 votes]: For a different perspective from the other answers: you may be interested in the notion of the (natural) density of a set of natural numbers, which offers an intuitive notion of 'size' that jibes with the idea that there are 'half as many' even numbers as whole numbers — but with the catch that very few sets of interest actually have a non-zero density!  The natural density $d(A)$ is defined as the limit $$\lim_{n\to\infty}\frac{\#\{i:i\leq n \wedge i\in A\}}{n}$$
- that is, intuitively, the limit of the percentage of numbers which are members of $A$, as we consider more and more integers.  It's easy to see that the natural density of the even numbers is $\frac12$, and in general the density of the multiples of $k$ is $\frac1k$, just as you would expect; but with one or two notable exceptions, most other sets of interest that have a density tend to have a density of $0$ (for instance, the primes, or the square numbers) and some sets (for instance, the set of 'numbers with an odd number of $1$s in their binary expansion') have no density at all because the limit doesn't exist.