Talk:Borel hierarchy

Plans

The plan is to expand this into a description of at least the boldface Borel hierarchy on a Polish space, including Sigma^0_a etc. But there is some doubt about how to deal with the lightface Borel sets -- do they go here, or in arithmetical hierarchy or somewhere else? CMummert 13:59, 13 June 2006 (UTC)

rank

Is the definition

The rank of a Borel set is the least $\alpha$ such that the set is in $\mathbf {\Sigma } _{\alpha }^{0}$.

really canonical? Do we have a reference? I could not find it in Kechris' book, nor in Moschovakis'. The definition

the least $\alpha$ such that the set is in $\mathbf {\Sigma } _{\alpha }^{0}\cup \mathbf {\Pi } _{\alpha }^{0}$

seems equally plausible. I have seen the expression "Borel set of finite rank" used, but at the moment cannot recall a place where (if ever) I have seen "Borel set of rank alpha".

--Aleph4 15:19, 1 April 2007 (UTC)

You may be right. I replaced the def with a def of "finite rank" which is less problematic and probably more relevant to the reader. CMummert · talk 19:20, 1 April 2007 (UTC)

Ill-stated definition

In the line

• A set $A$ is $\mathbf {\Sigma } _{\alpha }^{0}$ for $\alpha >1$ if and only if there is a sequence of sets $A_{1},A_{2},\ldots$ such that each $A_{i}$ is $\mathbf {\Pi } _{\alpha _{i}}^{0}$ for some $\alpha _{i}<\alpha$ and $A=\bigcup A_{i}$.

It is not evident from the definition that $\alpha$ is well-defined (or even bounded). A set $A$ could be the union of several different sequences of $A_{i}$ each producing a distinct $\alpha$.

Perhaps

• A set $A$ is $\mathbf {\Sigma } _{\alpha }^{0}$ for $\alpha >1$ if and only if $\alpha$ is the least integer such that there exists a sequence of sets $A_{1},A_{2},\ldots$ where each $A_{i}$ is $\mathbf {\Pi } _{\alpha _{i}}^{0}$ for some $\alpha _{i}<\alpha$ and $A=\bigcup A_{i}$.

-- Fuzzyeric (talk) 13:06, 19 November 2010 (UTC)

This is a feature rather than a bug. Every $\Sigma _{\alpha }^{0}$ set is also $\Sigma _{\beta }^{0}$ for every β > α. So rather than trying to divide up all the sets into disjoint pieces, we have a hierarchy of larger and larger classes of sets. — Carl (CBM · talk) 14:08, 19 November 2010 (UTC)

Definition of $\Delta _{\alpha }^{0}$?

The section on the lightface hiearchy needs a definition of $\Delta _{\alpha }^{0}$, but unless I'm missing something, no definition is given. Perhaps it just needs the line "A set is $\Delta _{\alpha }^{0}$ if and only if it is both $\Sigma _{\alpha }^{0}$ and $\Pi _{\alpha }^{0}$"? I don't know this area, I'm just guessing. Rahul Narain (talk) 17:07, 17 June 2014 (UTC)

This definition is still missing as of today. 67.198.37.16 (talk) 17:01, 27 November 2023 (UTC)