TITLE: What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?
QUESTION [8 upvotes]: Sometimes reading on wikipedia or in this site (and in very different context like topology, arithmetic and logic) I have found these symbols $\Sigma _n^i$,  $\Pi _n^i$ and $\Delta _n^i$. They are probably classes of something, but i don't know their name and i wasn't able to find something i can understand (I read about hierachies, but hierachies of what and, in which field of mathematics?)
I hope someone can give me an easy explaination of these things (classes?), how they are related, in which field(s) of mathematics these concepts appear and a formal definition (or a link).
Thanks in advance. and I apologize for errors (I'm using a translator).
Update
What is the meaning of the upper index $0$? In the logic use of this notation $\Sigma_n^1$ is the hieracy of the formulas in the language of second-order arithmetic. Intuitively I could think that the hierchies  $\Sigma _n^i$,  $\Pi _n^i$ and $\Delta _n^i$ are the hierachies of the formulas in the language of $(i−1)$-order arithmetic..probably I'm wrong but how this index is linked with other fields?

REPLY [13 votes]: A similar notation appears in

(In logic) Arithmetical hierarchy or Analytical hierarchy
(In descriptive set theory) Borel hierarchy or Projective hierarchy
(In computer science) Polynomial hierarchy

So it depends on context. Often (but not always) these symbols indicate that the hierarchy of object has the following diagram:

The meaning of the arrows depends on the context. For example, in the Borel hierarchy, $\bf\Sigma$ are families of sets closed under $\sigma$-union, $\bf \Pi$ are families closed under $\sigma$-intersection, $\bf\Delta$ are closed under both, and the arrows in the diagram indicate inclusion.