User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-1605058-20180130175328

\(\Pi^\alpha_\beta\) by itself doesn't refer to a theory, instead it just is a particular kind of formulas. In fact, this notation can mean two slightly different, but strongly related things, one in reference to set theory, one in reference to arithmetic. I will focus on the former.

\(\alpha\) in \(\Pi^\alpha\) refers to the order of a formula. To know what this means we have to realize that in general, in logic, we have different kinds of quantifiers. In set theory, the basic quantifiers are the ones which quantify over all sets (so we have \(\exists x:\) meaning "there is a set \(x\)" and \(\forall x:\) meaning "for all sets \(x\)"). We call those first-order quantifiers. Apart from those, we also have bounded quantifiers, which are those not ranging over all sets, but just elements of some fixed set \(A\): we have \(\exists x\in A:\) which means "there is an element \(x\) of \(A\)" and \(\forall x\in A:\) meaning "for all elements \(x\) of \(A\)".

With this, we can make sense of what \(\Pi^0_n\) and \(\Sigma^0_n\) mean, and those are defined inductively: a formula \(\varphi\) is called \(\Pi^0_0\) and \(\Sigma^0_0\) if it only uses bounded quantifiers. \(\varphi\) is called \(\Pi^0_{n+1}\) if it's of the form \(\exists x:\psi(x)\) for some \(\psi\) which is \(\Sigma^0_n\), and it's called \(\Sigma^0_{n+1}\) if it's of the form \(\forall x:\psi(x)\) for some \(\psi\) which is \(\Pi^0_n\). Hence a \(\Pi^0_n\) formula is of the form \(\forall x_1\exists x_2\dots\psi(x_1,\dots,x_n)\), with \(n\) alternating quantifiers. Note this is only defined for finite \(n\).

For higher orders, we consider higher-order quantifiers. The second-order quantifers, which I will denote by \(\exists^2,\forall^2\) (usually it's the variables which are denoted differently though, not the quantifiers themselves). Those quantify over families of sets, i.e. classes. So \(\exists^2 X:\) means "there is a class of sets \(X\)". From there the definition \(\Pi^1_n,\Sigma^1_n\) is very similar to the above, except for the base case, which is that \(\Pi^1_0,\Sigma^1_0\) refer to formulas only using first-order quantifiers, and in the general case we use n alternating second-order quantifiers.

Third-order quantifiers are ones which quantify over families of classes, and so on, from there defining all the \(\Pi^n_m\) is easy. For \(n\) we can even iterate transfinitely, but there are some technicalities I don't want to get into.

As for arithmetic, this is nearly all the same, except that the bounded quantifiers are now the ones of the form \(\exists m<n,\forall m<n\), and the first, second, third,... order quantifiers quantify over, respectively, natural numbers, sets of natural numbers, families of sets of natural numbers,... Apart from that, the definition of \(\Pi^n_m\) is the same.