Talk:Analytical hierarchy

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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 \Sigma^1_1 formula is a formula of the form \exists X \phi, where X is now a predicate and \phi \in \Sigma^1_0, while a \Sigma^1_1 set is a set of the form
\{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)

[edit] major problems seem to be fixed

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

[edit] 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.
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