EDIT: I posted an answer to this question. Can somone check?
Consider set $T_1,T_2,...T_p$ which are subsets of $\mathbb{Q}$
I want to create a new definition of "size" that distinguishes between subsets of $\mathbb{Q}$ that have "significantly" more elements inside of $\mathbb{Q}$ than other subsets. (Note that I want this size to apply to countably dense and countably finite subsets of $\mathbb{Q}$) .
For example, if we compared rational numbers to integers, we know there are significantly more rationals in rationals than integers in rationals. Morover, there are infinite rationals between every integer, further proving the previous statment.
Unfortunately, formal measures, which assigns a weight for each singleton $\left\{x\right\}$ in $T_p$,
$$\mu(T_p)=\sum_{x \in T_p}\mu(\left\{x\right\})$$
would not be meaningful since assigning zero or positive weight would give subsets of $\mathbb{Q}$ zero or infinite measure. This does not distinguish which of those subsets could have significantly more or less elements inside of $\mathbb{Q}$.
Cardinality is also a problem, since it counts the number of elements, rather than determine which subsets $\mathbb{Q}$ have more elements in $\mathbb{Q}$. Morover, the cardinality of any countably infinite set is infinity.
However, informal measures can be used. For example, the asymptotic density of a subset of $\mathbb{N}$ is the number elements the subset "fills" of natural numbers between $[0,b]$ as $b\to\infty$.
How can we create a new definition of "size" that constructs an informal measure of the subsets of $\mathbb{Q}$ and meets the following requirments?
-If $T_1=T_2$ and $\mu(T_1)=\mu(T_2)$
-If $T_1\subseteq T_2$ then $\mu(T_1) \le \mu(T_2)$