Talk:Analytical hierarchy

Mathematics B‑class Mid‑priority

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. B This article has been rated as B-class on Wikipedia's content assessment scale. Mid This article has been rated as Mid-priority on the project's priority scale.

Problem here -- the boldface/lightface distinction is not clearly made --Trovatore 7 July 2005 19:31 (UTC)

The term "analytical" refers exclusively to the lightface concept, whereas "projective" is a boldface notion. Therefore "projective set" should not redirect here. I'm planning a pointclass page where it might redirect instead. Also "analytic" and "co-analytic", which are boldface notions, should be removed from this page. --Trovatore 7 July 2005 20:35 (UTC)

---

The other major problem with this page is the conflation of formulas and sets. Consider for example the following passage:

A ${\displaystyle \Sigma _{1}^{1}}$ formula is a formula of the form ${\displaystyle \exists X\phi }$, where X is now a predicate and ${\displaystyle \phi \in \Sigma _{0}^{1}}$, while a ${\displaystyle \Sigma _{1}^{1}}$ set is a set of the form

${\displaystyle \{x:(\exists y\in S)\ R(x,y)\}}$,

where S is Borel and R is a relation.

First, the definition is incorrect, because no restriction is placed on the definability of R. But the problem that more exemplifies the difficulty with the page as a whole is that nothing is said about the underlying Polish space. --Trovatore 7 July 2005 20:48 (UTC)

major problems seem to be fixed

Latest comment: 19 years ago1 comment1 person in discussion

Accuracy tag removed (thanks to Ben Standeven). Tech tag removed; subject is inherently technical. --Trovatore 02:18, 20 September 2005 (UTC)Reply

edit on 2006-6-13

The main changes are:

• Carefully separate the cases of sets of numbers and sets of reals. This is an experiment that may be useful on arithmetical hierarchy as well.
• Emphasize the fact that this is lightface (maybe more emphasis needed?)
• I rephrased the introduction. The anlytical hierarchy is not really about second order logic; it is about higher type languages such as Z_2 or type theory in first order logic. Every use I know of is in the context of first order ZFC or first order Z_2.
• Add a reference to Rogers' book. I plan to add more references.

Hyperarithmetical

Latest comment: 18 years ago1 comment1 person in discussion

I've seen redlinks to 'hyperarithmetical heirachy' many places, and just recently noticed hyperarithmetical theory so I've created a redirect for now. It's possible that one or the other will need to be renamed in the future, but I figured this was a cheap way to get rid of a lot of dead links at least for now. --- all assuming that 'hyperarithmetical theory' is talking about the same thing... here's hoping! Zero sharp 20:46, 30 August 2007 (UTC)Reply

Removed sentence

Latest comment: 15 years ago1 comment1 person in discussion

"Note that it rarely makes sense to speak of a ${\displaystyle \Delta _{n}^{1}}$  formula; the first quantifier of a formula is either existential or universal."

Well, not really, the first quantifier of the normalized form, but not of the formula. As we just demonstrated any formula in ${\displaystyle \Pi _{n}^{1}}$  or ${\displaystyle \Sigma _{n}^{1}}$  is in ${\displaystyle \Delta _{m}^{1}}$  for all m>n. Rich Farmbrough, 16:23, 14 December 2009 (UTC).Reply