User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-30118230-20180130180340

$$\Pi^{n}_m$$ is a certain type of formula.

The definition is a bit formal but basically:

FOST sentences in the form $$\forall x,y$$ and $$\exists x,y$$ are called quantifiers,while sentences in the form $$\forall x\in y$$ and $$\exists x\in y$$ and are a specific type of quantifiers called bounded quantifiers.

We say that a formula $$\varphi$$ is both a $$\Pi_0$$ and a $$\Sigma_0$$ sentence if it has only bounded quantifiers.

$$$$ is a $$\Pi_{n+1}$$ sentence if $$\varphi$$ is a $$\Sigma_n$$ sentence and conversely,$$\Exists x: \varphi(x)$$ is a $$\Sigma_{n+1}$$ sentence if $$\varphi$$ is a $$\Pi_n$$ sentence.

(You can see here that formulae in the forms $$\Pi^{n}_m,\Sigma^{n}_m$$ only work for finite ordinals n,m)

Here $$\Pi_n,\Sigma_n$$ stand for $$\Pi^{0}_n,\Sigma^{0}_n$$ and $$\Pi^{n}_m,\Sigma^{n}_m$$ type formulae are a little more complicated but you get the basics here. These formula structures are usually combined with some axioms or schemas to make some theory $$T$$ more expressionate via assumptions about the statement in the axiom allowing more things to be decided in T.