# What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?

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?

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Perhaps you're thinking of the projective hierarchy? –  Nate Eldredge Mar 31 '13 at 14:12
$i+1$-order formulas, not $i-1$. $\Sigma^0_n$ is first-order, and $\Sigma^1_n$ is second-order. And so on. –  Asaf Karagila Apr 2 '13 at 21:11
Ah yes, thanks, i meant that even if I typed it wrong.@AsafKaragila –  MphLee Apr 2 '13 at 21:23

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.
Thanks, that makes all more clear to me. At least brings order in my mind. Anyways, I still don't understand what is the meaning of the upper index $0$, in the logic use of this notation $\Sigma^1_n$ is the hieracy of the formulas in the language of 2nd 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? –  MphLee Mar 31 '13 at 17:42