I have recently started learning/working with descriptive set theory. Several places, I've heard it defined as being the study of the "definability of the continuum" or something similar, but I'm having trouble understanding this definition.
As I understand it, if $\mathfrak{A}$ is a structure for a first order language $\mathcal{L}$, then a subset $S$ of the universe of $\mathfrak{A}$ is definable iff there is an $\mathcal{L}$-formula $\phi$ such that $x \in S$ if and only if $\mathfrak{A} \models \phi(x)$.
Based on this definition of definable, I would have expected the study of "the definability of the continuum" to begin by defining a language $\mathcal{L}$ and then studying those subsets $S \subseteq \mathbb{R}$ such that for some $\mathcal{L}$-formula $\phi$, $x \in S \iff \mathbb{R} \models \phi(x)$. This definitely does NOT seem to be what descriptive set theory is about.
To make matters worse, I've been told that in notations such as $\boldsymbol{\Sigma}^k_n$, $\boldsymbol{\Pi}^k_n$, etc. the top number is supposed to indicate quantifiers in $(k+1)^{th}$ order logic; the bottom number represents how many alternating blocks of $(k+1)^{th}$ order quantifiers there are; the $\Sigma, \Pi$ represents whether you start with an existential or universal block (respectively); and boldface vs. lightface represents whether or not parameters are allowed.
Again, this sounds to me like we're tallking about formulas in some language, but when I look at definitions like $\boldsymbol{\Sigma}^0_1 = $ open sets, I'm at a loss for what language that could be. I've tried and failed to come up with a language where the open subsets of an arbitrary Polish space are precisely the ones defined by $\Sigma_1$-formulas of the language (with parameters) and no book that I've checked seems to have much interest in defining $\boldsymbol{\Sigma}^0_1$ sets via definability in a structure.
So, what am I missing here? Is there some descriptive set theoretic definition of "definable" that I'm not aware of? If the definition of definable I gave above is correct, what is its connection to the (seemingly topological) properties like openness or $G_{\delta}$-ness tha descriptive set theory seems to be all about? Thank you for any help you can give me.
If your Polish space is $\omega^\omega$, then the language you are looking for simply has a basic predicate for each cylinder set $w\omega^\omega$ with $w \in \mathbb{N}^*$. This is of course a countably infinite language.
If we consider more general Polish spaces, then we fix some dense sequence $(a_n)_{n \in \mathbb{N}}$, and use as basic predicates $B(a_n,2^{-k})$ for $n,k \in \mathbb{N}$. Again, we are dealing with a countably infinite language here. However, now we need to be careful with negation: Since the complement of a basic open is no longer open, we either have to forbid negation altogether, or count any use of negation in determining the complexity of a formula.