Cardinality != Density?

Let $\mathbb{N}$ refer to the positive integers. Usually we define the density of a subset $A\subset \mathbb{N}$ with respect to the integers to be $$\lim_{N\rightarrow\infty}\frac{|\{m\in A:m\leq N\}|}{N}.$$ For example, the set of even numbers $$\{2,4,6,\dots\}$$ has density $\frac{1}{2}$. From this, it is not hard to see that for any $c\in[0,1]$ you can find a set with density $c$.

Notice that if $c>0$ this automatically implies the set has cardinality $\aleph_0$, but it is also possible to have a set of density $0$ with cardinality $\aleph_0$. For example, the set of powers of $2$, or the set of prime numbers.

Hope that helps,

I think that it is not completely clear to you what cardinality and density means.

Firstly cardinality is the "rawest" notion of size. For two sets to have the same cardinality they need only a bijection, which is to say "There exists a table with two columns, one for each set, and all the elements appear exactly once in that column."

The notion which you describe is called Dedekind infinite, which is to have the same cardinality as a proper subset.

As for density, this can be treated in several aspects:

• A dense ordered set is a set such that for every $x,y$ there is $z$ which is strictly between them. The rationals are such set, the positive integers are not.
  If there is a first and/or last point (for example all the non-negative rationals which are less or equal to one) then it is customary to exclude the end points from the density criteria and say that there is a first/last point as well.
• Topological dense sets: if $A$ is a topological space then $D\subseteq A$ is dense in $A$ if and only if its intersection with every open set is nonempty. The rationals are dense in the real numbers, in this sense, since every non empty interval contains a rational point inside it.

If the topological space is endowed with a linear order topology (e.g. the real numbers) then a dense set in the topological sense is a dense order as well (although not vice versa, consider all the real numbers versus the rationals between $-1$ to $1$)

The real numbers give example of a very large space (its cardinality is continuum) with a very small (countable) subset which is dense (topologically).

The third notion of size is measure, which roughly translates to volume. A large set is a set of positive measure (or equal measure to the measure of the entire space).

The Lebesgue measure is a way to determine the volume of subsets of the real numbers in the way we want it to work, that is if we just shift around the set it will not change its volume and if we stretch it the volume will increase as the factor we stretched by.

One can build a set which is of Lebesgue measure $0$ (i.e. has no volume at all), not dense at any point (that is if a point is outside the set then it has an open interval which does not intersect this set) and yet it is of the cardinality of the continuum. This can be generalized to have any volume that we want as well.

Therefore the notions are hardly related, if we have a very small (in cardinality) set which is dense (large topologically) and another set which is very large in cardinality but topologically speaking is very very small.

To sum up, there are many different ways to measure how big a set is and it gets more and more complex with every new technique you acquire (filters, cofinality, and more). They may or may not be related or correlated, but the case is usually that we want a new way of "sizing" up sets, mostly because the ones we have are insufficient or cumbersome for the task at hand.

Subsets of the naturals can have different natural densties. If the natural density is greater than zero, the subset is countably infinite. Think of all the naturals (density 1) and the even numbers (density 1/2). But there are infinite subsets with natural density zero and infinite subsets for which the natural density cannot be defined.