Talk:Borel hierarchy

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) [reply ]

Is the definition

The rank of a Borel set is the least ${\displaystyle \alpha }${\displaystyle \alpha } such that the set is in ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}${\displaystyle \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 ${\displaystyle \alpha }${\displaystyle \alpha } such that the set is in ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}\cup \mathbf {\Pi } _{\alpha }^{0}}${\displaystyle \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) [reply ]

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) [reply ]

In the line

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

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

Perhaps

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

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

This is a feature rather than a bug. Every ${\displaystyle \Sigma _{\alpha }^{0}}${\displaystyle \Sigma _{\alpha }^{0}} set is also ${\displaystyle \Sigma _{\beta }^{0}}${\displaystyle \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) [reply ]

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

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

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