Difference between propositional language and set of all formulas

[V0ODO0CH1LD]

I am currently reading Rautenberg's book on mathematical logic, in it he defines a propositional language $\mathcal{F}$, set theoretically, as the smallest (i.e. the intersection) of all sets of strings $S$ built from propositional variables ($\ p_1,p_2,\ldots$) as well as any binary connectives on those variables, with the properties:
$$(1)\ p_1,p_2,\ldots\in{}S;\quad{}(2)\ \alpha,\beta\in{}S\Rightarrow(\alpha\circ\beta)\in{}S$$
How does this definition work? Is $S$ supposed to be representing strings or a set? Is $S$ supposed to be the set of all strings with those properties? Or is $\mathcal{F}$? Also, what am I supposed to be intersecting there?

 

[gopher_p]

I am currently reading Rautenberg's book on mathematical logic, in it he defines a propositional language $\mathcal{F}$, set theoretically, as the smallest (i.e. the intersection) of all sets of strings $S$ built from propositional variables ($\ p_1,p_2,\ldots$) as well as any binary connectives on those variables, with the properties:
$$(1)\ p_1,p_2,\ldots\in{}S;\quad{}(2)\ \alpha,\beta\in{}S\Rightarrow(\alpha\circ\beta)\in{}S$$
How does this definition work? Is $S$ supposed to be representing strings or a set? Is $S$ supposed to be the set of all strings with those properties? Or is $\mathcal{F}$? Also, what am I supposed to be intersecting there?

Not sure what you mean by the definition "working".

$S$ is an arbitrary set of strings which has the two properties listed. The properties listed are properties of sets of strings, not properties of strings. $\mathcal{F}$ is a particular set of strings which has the two properties listed; the smallest such set in the sense that it is contained in all other such sets. The intersection is taking place over the family of all sets satisfying the two properties.

For comparison, one slightly shady way of defining the natural numbers is to say that $\mathbb{N}$ is the smallest of all sets $S$ of members of $\mathbb{R}$ with the properties:
$$(1)\ 0\in S\text{ and }(2)\ x\in S\Rightarrow x+1\in S$$