Which mathematics theory studies structures like these? Let $A_p$ be the set of all numbers whose prime factors are all in first $p$ prime numbers.
example: $A_2= \{2,3,4,6,9,12,16,18,\ldots \}$ (all of these numbers can be generated by repeatedly multiplying only $2$ and $3$ the first two prime numbers).
as $p \to \infty$
intuition is that this set would cover all of natural numbers.
Would that make this $(A_\infty)$ set equivalent in some sense to $\mathbb N$ itself. Is a theorem proved on $\mathbb N$ imply that it is also true on the set that I defined (and vice versa).
Is there any branch of mathematics that deals with these kind of structures.
Is there a specific theorem that specifically deals with the above mentioned theorem.
Thanks is advance ,sorry for my poor mathematical literary skills.
 A: You haven't really defined a meaning for your $A_\infty$, unless you think "the first infinity prime numbers" make sense.
If you choose a definition for $A_\infty$ -- such as the union of all $A_p$ for finite $p$,
$$ A_\infty = \bigcup_{p\in\mathbb N} A_p $$
then it will be easy to show that $A_\infty=\mathbb N$. (Note that you really ought to let $1$ be an element of each of your $A_p$s. Since $1$ has no prime factors, in particular it has no prime factors outside the first $p$ prime numbers, and you can generate it by multiplying together none of the first $p$ primes).
(I'm also assuming for simplicity that $0$ does not count as a "natural number" for you).
But there's no general theory that says that if you have defined some objects $B_n$ for $n\in\mathbb N$, then the notation $B_\infty$ must mean such-and-such. That's a definition you have to decide on in each particular case.
Mathematically, it would be perfectly valid to define $A_\infty$ to mean the set $\{42,117\}$ -- there is no automatic relation between $A$ with a number as a subscript and $A$ with an "$\infty$" symbol as a subscript. The worst that can happen is that your readers will be confused.
A: I would be more comfortable calling it $A_n$ than $A_p$ because the letter $p$ is often used to refer only to a prime number.
Probably the concept you need is that of a union of sets, and that is not defined by saying $n\to\infty$.  The union
$$
\bigcup_{n\in\mathbb N} A_n \tag 1
$$
is defined by saying that a number is a member of the union if and only if for at least one $n\in\mathbb N$, that number is in $A_n$.  There is no mention of limits as anything expressed as $n\to\infty$.
This union is in fact all of $\mathbb N$ for a simple reason: every member $k$ of $\mathbb N$ is divisible by finitely many prime numbers, so there is some $n$ such that $k\in A_n$.
One way of proving that infinitely many prime numbers exist is by showing that
$$
\sum_{m\in A_n} \frac 1 m <\infty
$$
so that $A_n$ cannot be all of $\mathbb N$, since $\displaystyle\sum_{n\in\mathbb N} \frac 1 n =\infty$.  It is for that reason that you cannot just take a union of only finitely many $A_n$ and get all of $\mathbb N$.
