# 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?

• 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

A similar notation appears in

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.

• 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
• I don't know the answer, but this book looks promising: Kechris, A. S., Classical Descriptive Set Theory – Yoni Rozenshein Mar 31 '13 at 18:47
• Thanks. Well I'll search an online version of this book, meanwhile i'll update my question. – MphLee Mar 31 '13 at 18:54