I'm Confused About Size, Combinatorial Classes and Counting Sequences My Teacher tried to explain these concepts like so:

My Problem
I'm quite confused by the definition given. How can "the size" which is a function mapped from our countable set A to all the non-negative integers, be a non-negative integer? Aren't numbers and functions two distinct mathematical objects? I'm also pretty confused by what is meant by "If for every $n$ there are finitely many elements of A of size n then we call A a combinatorial class" could someone explain to me what is combinatorial class?
 A: The definition uses the word size for two different things. First, each element of $\mathcal{A}$ has an attribute size that is a non-negative integer. For instance, if $\mathcal{A}=\Bbb Z^+$, we might define the size of of $n\in\mathcal{A}$ to be the number of distinct prime factors of $n$. In that case the sizes of $2,3,4$, and $5$ would all be $1$, but the size of $6$ would be $2$, and the size of $1$ would be $0$. This attribute gives us a size function from $\mathcal{A}$ to the set of non-negative integers, a function that assigns to each $a\in\mathcal{A}$ its size. The definition that you’ve been given calls this size function simply the size rather than the size function. As a (slightly confusing) result, the word size is used both for the function and for its values. You’re absolutely right in thinking that these are different things; the author of that definition has simply chosen to use the same word for both.
If we have a countable family $\mathcal{A}$ equipped with a size function that assigns to each $a\in\mathcal{A}$ a non-negative integer $|a|$, called the size of $a$, we can ask how many elements of $\mathcal{A}$ have size $0$, how many have size $1$, and so on. If $n$ is a non-negative integer, let $\mathcal{A}_n=\{a\in\mathcal{A}:|a|=n\}$ be the set of $a\in\mathcal{A}$ of size $n$. If all of the sets $\mathcal{A}_n$ are finite, we call $\mathcal{A}$ a combinatorial class.
The family $\mathcal{A}$ in my example is not a combinatorial class: $\mathcal{A}_0=\{1\}$ is finite, because $1$ is the only positive integer with $0$ distinct prime factors, but $\mathcal{A}_1$ is infinite: it includes all primes, and positive powers of primes. 
If $\mathcal{A}$ is a combinatorial class, for each non-negative integer $n$ we let $a_n$ be the number of elements of size $n$, i.e., the number of elements of the set $\mathcal{A}_n$: since we’re dealing with a combinatorial class, each $a_n$ is a non-negative integer. It can be $0$: there might not be any elements of $\mathcal{A}$ of size $3$, say, in which case $a_3=0$. Finally, the sequence $\langle a_n:n\in\Bbb N\rangle=\langle a_0,a_1,a_2,\ldots\rangle$ is called the counting sequence of the combinatorial class $\mathcal{A}$.
