Talk:Field of sets
|WikiProject Mathematics||(Rated Start-class, Low-importance)|
Made an own page for this as fields of sets are much wider than just sigma algebras.
Have been busy on other pages but will add more to this page soon. - 13 Oct 2004
Will convert to LaTeX, looks ugly without it. Kuratowski's Ghost 23:16, 11 Nov 2004 (UTC)
Began the painful Texification process, bear with me. Lot more to say in this article but not finding as much time as I'd like. Kuratowski's Ghost 15:01, 17 Nov 2004 (UTC)
Currently too much ugly formal notation, will try rewording to avoid notation. Kuratowski's Ghost 17:19, 30 Nov 2004 (UTC)
Said all I have to say for now, if anyone knows of any other interesting uses or aspects of fields of sets, please add. Kuratowski's Ghost 16:46, 24 Dec 2004 (UTC)
Isn't the "subset of the powerset of X" just a 'subset of X'? 14:51, 10 Jul 2007 (UTC)
- No Kuratowski's Ghost 22:35, 10 July 2007 (UTC)
Suggesting a simpler language
Some passages need rewording to become intelligible for an audience that does not already know. At the moment I have in mind the statements
Every finite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element).
(1) What is a whole power set?
(2) Whose set of atoms? (what does "it" refer to in "its set of atoms"?) Do Boolean algebras have atoms?
(3) Below what? (what does "it" refer to in "the set of atoms below it"?) Do elements of Boolean algebras have atoms below them?
(4) What means "below"? I don't know about lattices and their graphical representations. Nobody reminded me to think of subsetness as a partial ordering.
(5) What is a join? I came to this article searching for a definition of "an algebra", and most other wiki pages I found seemed wrong since they speak mostly about bits and bit strings - not to mention the pages about 'algebra' in general.
The statement makes perfect sense once the general mind frame is clear, but it is not very helpfull to the reader that lacks this frame. The sheer number of questions that arise in a reader's mind makes it hopeless to even start guessing. Perhaps part of the problem is a mismatch between the pages under boolean algebra and this page. I suggest we try to keep most pages as self-contained as possible, and include enough words of introduction of the concepts used, or, when available on wikipedia, links.
The page does contain links already, and perusing each does allow the reader to dechiffer the text. However, I would like to encourage contributors to make this task less ardous, even if that means repeating some stuff in a number of pages. There could also be a link that is singled out early in the page as the one leading to an exposition of the terminology used in the page.
Let me attempt a rephrasing of the quote above.
While one often arrives at a particular boolean algebra by considering a certain selection A of the possible subsets of some set X, if A is finite it is always possible to find a set Y to replace X such that the same boolean algebra can be represented by the set of all subsets of Y. In terms of order theory, the members of Y correspond to the atoms of A.
What do you think? In this wording, I am trying to reduce the number of underlying concepts the reader needs to know about. Also, I try to make it optional to bother about the order theory and atoms, while still offering that perspective.
It is true that seeing cross connections between the various fields of mathematics is part of the fun, but reading wikepedia math articles has become an almost impossible task because every link one follows trying to get a foundation of the terminology used in a first page, creates the need to follow three more links to understand the terminology in the new page. I hope contributors will eventually find time to take most articles down to a simpler level.
I am not implying that just fixing the above passus will fix the page. What is the deal with the Stone representation? Why is a representation needed? The whole treatment lacks all forms of motivational stuff. Making wikipedia's math pages a good source for half-educated readers is a monumental task. I suppose that in the first years it has been more beneficial for the public that contributors quickly wrote about many subjects, but we need that the structure be cleaned up eventually, and I am hoping to give another push in the right direction.
Mathematics is generally about abstractions and generalizations, and so even a concept like a Boolean algebra is treated as an abstract concept. However, an encyclopedia article may be the wrong place to achieve the maximum abstraction and generality. The reader needs a simple model, and needs relate as much as possible of the subject to this simple model, before he is ready to see the subject in a thousand guises and models. While some concepts warrant separate introductory and advanced treatment, I believe most articles should have a reasonably pedagogical structure where the generailzations and alternative models are introduced toward the end. Readers at a higher level can skip sections 2 through n. They will probably discover that the articles have this structure. PerezTerron (talk) 05:09, 29 January 2008 (UTC)
Hwo invented the terminology?
finite vs. countable
A glaring problem in the very first sentences is a failure to distinguish finite vs countable number of operations. This propagates to the section on topology: in a topology, one is allowed countable unions, finite intersections, and there is no talk of complements, because that would ruin things for a topology (the complement of an open set is not open; adding complements gets you Borel sets, instead, which are something different). So there is some kind of creeping incorrectness hard at work in the article :-( linas (talk) 16:09, 21 September 2012 (UTC)
- I have got to stop skimming articles and coming to knee-jerk reactions. Still, the lack of discussion of cardinality is vaguely un-nerving, as this is usually important in many other areas, and so seemingly should matter here as well. linas (talk) 16:14, 21 September 2012 (UTC)
Basic examples missing
It's also missing the basic infinite examples (before one gets to extra structure) like finite-cofinite or intervals, which one can even find in Springer EOM or in Givant and Halmos Introduction to Boolean Algebras etc. 22.214.171.124 (talk) 21:00, 10 April 2015 (UTC)