Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
Perhaps you're thinking of the projective hierarchy? –  Nate Eldredge Mar 31 '13 at 14:12
1  
$i+1$-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

1 Answer 1

up vote 10 down vote accepted

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:

Hierarchial 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.

share|improve this answer
    
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
1  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.