Talk:Field of sets

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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&#39;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)


 * Well, I just rewrote this paragraph; hopefully, more accessible now. Boris Tsirelson (talk) 08:43, 16 November 2019 (UTC)

Hwo invented the terminology?
I'm curious because Mac Lane and Birkhoff seem to disapprove it in a way when they put field in scare quotes. Tijfo098 (talk) 01:05, 15 April 2011 (UTC)

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
One obvious question is if there are finite fields of sets other than the powerset etc. 86.127.138.67 (talk) 17:05, 10 April 2015 (UTC)

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. 86.127.138.67 (talk) 21:00, 10 April 2015 (UTC)

Confusing
This article is deeply confusing. Aside from the multiple unanswered, orphaned comments above, I also see stuff like this:
 * Given a field of sets $$\mathbf{X}= \langle X, \mathcal{F} \rangle$$ the complexes form a base for a topology, we denote the corresponding topological space by $$T(\mathbf{X})$$. Then $$T(\mathbf{X})$$ is always a zero-dimensional space.

Hmm. I guess this is perhaps the fine topology, or some discrete topology?


 * There isn't anything wrong in the article, the problem is that you do not understand what the term "base for a topology" means. The article does link the Wikipedia article explaining it, I don't think we need to explain standard topology terms again in this article.


 * I thought I understood what a base was. I don't understand how we got to "zero-dimension". Is zero-dimension referring to the Lebesgue covering dimension? If so, then I guess this article wants me to think of the power set of $$X$$ as a totally disconnected set? In that case, I guess I agree: if I think of only the points, the atoms of $$X$$, then, yes, the power set $$2^X$$ is totally disconnected, no matter what the cardinality of $$X$$ is. This is conventionally called the box topology, so I guess you want me to think of $$2^X$$ endowed with the box topology, which would make it zero-dimensional. But certainly, there are subsets of $$2^X$$ that can be endowed with topologies that have Lebesgue covering dimension greater than zero.


 * For example, I can take $$X=\mathbb{N}$$ and the power set is the Cantor space. Let me remove from the Cantor space all strings that terminate with an infinite sequence of 1's. The result is a field of sets, I think: closed under complements, finite unions and finite intersections. There is no way to create a point with an infinite run of all-one's with only a finite number of intersections and unions. (right? or am I hallucinating?) The definition of "field of sets" says "closed under the intersection of pairs of sets", not "closed under countable intersections". So this construction meets the definition of a field of sets. The points of this field can be placed in one-to-one correspondence with the reals, because I removed the double-counting of the dyadic rationals. The natural topology on the reals has Lebesgue covering dimension of one, so I've exhibited a field of sets that is a proper subset of the Cantor set that has dimension one, not zero. So again, where did this dimension-zero thing come from?

Enquiring minds want to know. Because later on, this very same article states:
 * If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebra and the corresponding field of sets is called a measurable space. The complexes of a measurable space are called measurable sets.

Great! Except that measurable spaces are typically not zero-dimensional.


 * The article does not state that a topological space that also happens to be a measureable space, is zero-dimensional. It says the topology generated by the complexes (measurable sets) is zero-dimensional. The only one confused is you.


 * That is certainly NOT what it says! It says that ALL fields of sets are zero-dimensional. It doesn't say "some of them".

So what happened? I guess we are now talking about the same field of sets but using a different topology? Or perhaps we threw away some of the complexes along the way, so that its a different "field of sets", but built on the same set X? ?? By the way, does it matter when X is countable, vs. when it isn't? Boolean algebras for finite, countable and uncountable sets are rather different from one-another. What happens if I do forcing (mathematics) and insert some additional sets into the complex of the field of sets? Somehow, this is all very confusing. I'm looking for precise, exact definitions, and they're not here. I'm not even vaguely expert enough to fix this without adding additional errors. 67.198.37.16 (talk) 18:58, 8 March 2018 (UTC)


 * I just added an "expert attention" tag to the article. Pretty much every issue raised in all of the different comments above needs to be addressed and resolved. 67.198.37.16 (talk) 19:07, 8 March 2018 (UTC)


 * Yes, we are talking about the same field of sets but using different topologies. This is usual enough. And I do not agree that "measurable spaces are typically not zero-dimensional". There are a number of (not always equivalent) definitions of dimension for topological space, but no one defines dimension for measurable spaces. In fact, all uncountable Borel sets in Euclidean spaces of all (finite) dimensions are isomorphic as measurable spaces! Boris Tsirelson (talk) 09:36, 16 November 2019 (UTC)
 * A clarification just added to the article: "...just one of notable topologies on the given set of points..." Boris Tsirelson (talk) 09:48, 16 November 2019 (UTC)
 * Is expert attention still needed? Boris Tsirelson (talk) 09:52, 16 November 2019 (UTC)
 * Yes. 67.198.37.16 (talk) 01:55, 4 October 2020 (UTC)

Definition
I corrected the incorrect definition: X must be an element of F, that is, an algebra F' over a strict subset Y of X is not define an algebra over X (unless we add $$Y^c$$ to the set of atoms). I also wrote intersections as consequences, not definition, both in the introduction and in the sigma algebra section. For example, Rudin's Real Analysis defines sigma algebras by requiring complements and countable unions, mentioning intersections as a consequence (by the infinite variants of De Morgan's laws). Could somebody provide a similar link to the definition of an algebra? This general habit of formally minimizing definitions is important, so that the user does not have to verify too many things. However, I wrote $$X\in\mathcal F$$, not the equivalent $$\emptyset\in\mathcal F$$, so as to make it absolutely clear and crisp that F' is not an algebra over X. Algebra of sets (Enc. of Math) proves that I am correct in requiring $$X\in\mathcal F$$ but uses the empty set and even formally requires intersections, so it is not a good reference for the definition section. I added it to "External links" anyway. Preferably there would also be a new section "Definition" that would be very clear and precise. --Rigmat (talk) 11:43, 13 April 2020 (UTC)

Actually, $$X\in\mathcal F$$ could be replaced by the formally (but not truly strictly) weaker "F is nonempty", as X equals the union of any complex with its complement. But I still prefer $$X\in\mathcal F$$, to make that fact crisp and clear. --Rigmat (talk) 13:59, 13 April 2020 (UTC)

"Complex algebra" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Complex algebra and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 July 1 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. D.Lazard (talk) 19:44, 1 July 2022 (UTC)