User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-27513631-20180130205652

I'll try to give a more discussion-relevant answer.

As LittlePeng and Boboris have explained, \(\Pi^n_m\) etc. refer to the complexity of formulae. For instance:

A \(\Pi^0_0\) formula refers to one where all of the quantifiers are bounded. This is the same as \(\Sigma^0_0\) and \(\Delta^0_0\).

A \(\Pi^0_1\) formula is of the form \(\forall x,\cdots,z \in N(\psi(x,\cdots,z))\), where \(\psi(x,\cdots,z)\) is a \(\Sigma^0_0\) formula.

A \(\Pi^0_2\) formula is of the form \(\forall w,\cdots,x \in N\exists y,\cdots,z \in N(\psi(w,\cdots,z))\), where \(\psi(w,\cdots,z)\) is a \(\Pi^0_0\) formula.

\(\Pi^1\) and \(\Sigma^1\) formulae are defined in much the same way, but using \(x \subseteq N\) (equivalently, \(x \in \mathcal P(N)\)). Here, \(N\) is normally the set of natural numbers but is occasionally a larger set, such as \(V_\kappa\) when defining weakly compact cardinals.

However, (despite what LittlePeng seems to suggest) it is very uncommon to see \(\Pi^1\) or \(\Sigma^1\) formulae when \(N\) is the entire set-theoretic universe. That's why you'll just see \(\Pi_n\) or \(\Sigma_n\) instead: the \(\forall x \in N\) and \(\exists x \in N\) become \(\forall x\) and \(\exists x\) respectively.

This relates to the strength of functions in one main way: how much of a theory is needed to make that function make sense? One of the simplest ways of restricting this is through quantifiers: \(\Pi^1_2-CA_0\) corresponds to normal second-order arithmetic but restricted so that, instead of being able to make sets of natural numbers from arbitrary predicates, you can only use \(\Pi^1_2\) predicates.

You might find things like \(\Delta^1_2-CA\) or \(\Sigma^1_4-DC\) on similar places in the wiki: they mean the restriction of some part of second-order arithmetic to allow use of fewer formulae.

The uses of \(\Sigma_1\) above are a bit more complex, but basically in KP we lack a similar comprehension principle (full seperation) that we have in set theory, and this can be described either directly, by adding seperation for \(\Sigma_1\) formulae, or saying that there are sets which are hard to define using just smaller sets and \(\Sigma_1\) formulae inside some specific sets \(L_\kappa\), which is a much more intricate definition to state formally.