Talk:Prewellordering
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Untitled
[edit]This is leading up to the definition of the prewellordering property and its consequences for pointclasses that have it (reduction for the pointclass, separation for the dual pointclass). At a minimum, there first needs to be a pointclass page.
Fairly complete now--main problem is the redlinks. The pointclass page I want to put together would have considerable overlap with analytical hierarchy, but the latter is seriously flawed by a lack of a lightface/boldface distinction, and by its conflation of formulas with pointsets (this conflation is fairly routine in a context where it's understood, but it isn't sufficiently explained on the existing page) --Trovatore 7 July 2005 20:01 (UTC)
Image request
[edit]need pics for Reduction and Separation properties. For the latter I don't really want an analogue of Moschovakis' Figure 4B1; I'd rather have one that shows separation of two disjoint sets in Γ. --Trovatore 16:27, 7 September 2005 (UTC)
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Connected=total?
[edit]The link connected does not make clear what connected means here. Does it mean a total relation?--Patrick 09:20, 17 June 2007 (UTC)
Apparently it does, I changed it.--Patrick 09:48, 17 June 2007 (UTC)
Jochen's diagram
[edit]
Jochen Burghardt added the image at right, then removed it, saying it was "not wellfounded due to cycles". But it's only the strict part that needs to be wellfounded. Unless I'm making a silly mistake, this is indeed an example of a prewellordering.
That said, I'm a little undecided whether I think it's an example we should put in the article. For the most part, in practice, we consider prewellorderings on some nontrivial Polish space (the natural numbers are a trivial Polish space). A more canonical example would be the one that says, if two real numbers both code countable ordinals, then the one that comes first is the one that codes the smaller ordinal, and the ones that don't code any ordinal come after all of those. That's a little harder to put in a simple image. Still, Jochen's image is actually kind of nice; it does show the basic idea. --Trovatore (talk) 21:31, 26 April 2020 (UTC)
- @Trovatore:: I have made several mistakes: When I saw this page listed at Category:Wikipedia_requested_mathematical_diagrams, I glanced at the article, thought about a diagram, and came up with the one shown here. After that, when I intended to delete the diagram request on this talk page, I learned that a completely different diagram has been requested. I don't have sufficient detail knowledge to design a diagram that matches your request, so I kept my example diagram, being better than nothing. — Yesterday, I confused myself by not considering that Hasse diagrams are understood to denote the transitive closure of what they actually depict, so I thought the strict-part requirement would only rule out cycles of length ≤2.
- To sum up, it was me who made not one silly mistake, but two of them. Sorry for that! If you like, undo my self-revert to show the image, until a better one is available. If the Polish space article contained a formal definition in elementary terms, I'd try to understand it and to design an illustration of your canonical example, but it seems I'd have to learn half of a topology textbook before I can understand Polish spaces. - Jochen Burghardt (talk) 10:48, 27 April 2020 (UTC)