Talk:Analytical hierarchy

Mathematics B‑class Mid‑priority

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boldface distinction

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

alternating

Latest comment: 15 years ago2 comments2 people in discussion

MAJOR PROBLEM here -- the description of the the abstract levels of the hierarchy do not adequately reflect the alternating nature of the quantifiers. In fact, no where on the page is the word "alternating" found. See the planet math article for a correct description. —Preceding unsigned comment added by 131.107.0.98 (talk) 18:44, 24 June 2010 (UTC)Reply

It is true that the page doesn't use the word "alternating", but I think the definition here is correct. Is there something in particular wrong with it? — Carl (CBM · talk) 00:26, 25 June 2010 (UTC)Reply