Talk:Projective hierarchy

comments
This used to be a redirect to analytical hierarchy, but that doesn't make any sense as "analytical" is a lightface notion, whereas "projective" is boldface. This page and analytic set are candidates for a future merge into the pointclass page, when I get that written. --Trovatore 8 July 2005 06:19 (UTC)

Requested move
Projective set → Projective hierarchy

There's no sense in having both articles, and the "hierarchy" title better reflects the content. --Trovatore 06:56, 31 March 2007 (UTC)


 * Done. CMummert · talk 14:02, 31 March 2007 (UTC)


 * I think it is nice to have a separation between X hierarchy and X set. For example:
 * Arithmetical hierarchy / Arithmetical set
 * Analytical hierarchy / Analytic set
 * Borel hierarchy / Borel set (= Borel algebra)


 * There is a little duplication of content, but I think it is helpful to a naive reader to start with the non-hierarchy definition and later learn about the stratification. CMummert · talk 14:06, 31 March 2007 (UTC)
 * Well, you can make a case for that, but it does make maintenance and improvement more difficult. (By the way the "analytical hierarchy/analytic set" juxtaposition is wrong.) --Trovatore 16:37, 31 March 2007 (UTC)


 * The Lightface and darkface page is still unwritten. Not being a descriptive set theorist, I tend mentally identify the corresponding hierarchies.CMummert · talk 17:18, 31 March 2007 (UTC)
 * There's a pointclass page that treats that material, with redirects from lightface, lightface pointclass, boldface pointclass, and a link from boldface (disambiguation). No one seems to have touched that page but me. I think it's a critical concept, given that it's the essential subject matter of descriptive set theory (one could almost say it should bear the same relation to the descriptive set theory article that set bears to set theory). I think I did a decent start-class job on the article, but I wonder whether people are actually using the material, given that no one has edited it. --Trovatore 07:11, 1 April 2007 (UTC)
 * And thanks for reverting me at Analytic set, I remembed the distinction when I added it to analytical hierarchy but not this morning. I wasn't thinking.  CMummert · talk 18:27, 31 March 2007 (UTC)

Relationship to the Borel hierarchy?
We seem to have almost the same content on the page for the Borel hierarchy and if one blurs one's eyes, they cannot be told apart. Perhaps some clarifying distinction should be drawn. Well, I mean, the Borel hierarchy starts with $$\mathbf{\Sigma}^0_1$$ and the analytic hierarchy does not show up till much later, as $$\mathbf{\Sigma}^1_1$$, but it seems that perhaps this should be pointed out in the opening paragraphs. 67.198.37.16 (talk) 20:28, 27 November 2023 (UTC)

More information need to be mentioned in the article
For example, an important fact from the German version: "Alle Klassen $$\Sigma^1_n, \Pi^1_n$$ und $$\Delta^1_n$$ sind abgeschlossen bezüglich abzählbarer Durchschnitte und abzählbarer Vereinigungen, insbesondere ist $$\Delta^1_n$$ eine σ-Algebra. " (The German version of the pages uses the lightface symbols for both Borel hierarchy and projective hierarchy. ) If I understand it correctly, it says that families $$\mathbf{\Sigma}^1_n$$ and $$\mathbf{\Pi}^1_n$$ are all closed under countable union and intersection, so $$\mathbf{\Delta}^1_n$$ is a σ-algebra.

We know that $$\mathbf{\Sigma}^1_1$$ is closed under countable union and intersection from the page analytic set. Since the image of union is the union of images, the implication "$$\mathbf{\Sigma}^1_n$$ is closed under countable intersection $$\Rightarrow$$ $$\mathbf{\Sigma}^1_{n+1}$$ is closed under countable union" would be trivial, but to prove that $$\mathbf{\Sigma}^1_n$$ is closed under countable intersection seems to be nontrivial for me. A proof would be well appreciated.

Another example: the diagram in the German page says that $$\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n\subset\mathbf{\Delta}^1_{n+1}$$, which is equivalent to $$\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n\subset\mathbf{\Sigma}^1_{n+1}$$. This means that every $$\mathbf{\Sigma}^1_n$$ set and every $$\mathbf{\Pi}^1_n$$ is the projection of a $$\mathbf{\Pi}^1_n$$ set; while the latter implication is trivially true, I have no idea about the former even with n = 1. The inclusion gives us the inclusion of σ-algebras $$\mathbf{\Delta}^1_1\subset\sigma(\mathbf{\Sigma}^1_1)\subset\mathbf{\Delta}^1_2\subset\cdots\subset\mathbf{P}$$. Perhaps in fact each inclusion is strict in eveyy uncountable Polish space? (We know that this is true for the first inclusion as stated here, of course some choice may be needed; this post may have addressed the second, although I don't know if it would work for eveyy uncountable Polish space.)

What's more, the inclusion $$\mathbf{\Pi}^1_n\subset\mathbf{\Delta}^1_{n+1}$$ tells us that $$\mathbf{\Sigma}^1_n$$ sets are precisely the projections of $$\mathbf{\Delta}^1_n$$ sets, so $$\mathbf{\Delta}^1_n$$ sets are precisely those sets such that themselves as well as their projections are all $$\mathbf{\Delta}^1_n$$ sets: by definition $$\mathbf{\Delta}^1_n\subset\mathbf{\Sigma}^1_n$$, and the projections of $$\mathbf{\Sigma}^1_n$$ sets are also in $$\mathbf{\Sigma}^1_n$$. Conversely, a $$\mathbf{\Sigma}^1_n$$ set is projection of $$\mathbf{\Pi}^1_{n-1}$$ set, and the latter is a $$\mathbf{\Delta}^1_n$$ set. (This works for n ≥ 2; for n = 1, the proposition "every analytic set is the projection of a $$\mathbf{\Delta}^1_1$$ set" is something outside the hierarchy: we have to show that every analytic set is the projection of a Borel set, and $$\mathbf{\Delta}^1_1$$ sets are precisely Borel sets (Suslin's theorem).) 129.104.241.214 (talk) 06:07, 12 February 2024 (UTC)