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 $$\mathbf{\Sigma}^0_\alpha$$.

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 $$\mathbf{\Sigma}^0_\alpha\cup \mathbf \Pi^0_\alpha$$

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 $$\mathbf{\Sigma}^0_\alpha$$ for $$\alpha > 1$$ if and only if there is a sequence of sets $$A_1,A_2,\ldots$$ such that each $$A_i$$ is $$\mathbf{\Pi}^0_{\alpha_i}$$ 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 -- Fuzzyeric (talk) 13:06, 19 November 2010 (UTC)
 * A set $$A$$ is $$\mathbf{\Sigma}^0_\alpha$$ 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 $$\mathbf{\Pi}^0_{\alpha_i}$$ for some $$\alpha_i < \alpha$$ and $$ A = \bigcup A_i$$.


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

Definition of $$\Delta^0_\alpha$$?
The section on the lightface hiearchy needs a definition of $$\Delta^0_\alpha$$, but unless I'm missing something, no definition is given. Perhaps it just needs the line "A set is $$\Delta^0_\alpha$$ if and only if it is both $$\Sigma^0_\alpha$$ and $$\Pi^0_\alpha$$"? 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)