Talk:Stone's representation theorem for Boolean algebras
From Wikipedia, the free encyclopedia
 |
This article is within the scope of the WikiProject Philosophy, which collaborates on articles related to philosophy. To participate, you can edit this article or visit the project page for more details. | ||||||||
| ??? | This article has not yet received a rating on the quality scale. | ||||||||
| ??? | This article has not yet received an importance rating on the importance scale. | ||||||||
|
|||||||||
Talk moved from former "Stone's duality". --Markus Krötzsch 14:39, 18 Apr 2004 (UTC)
- Stone's duality generalises to infinite sets of propositions the use of truth tables to characterise elements of finite Boolean algebras.
I think of a truth table as a device to define logical operators, i.e. to define functions {T,F}n→{T,F}, and of the elements of a Boolean algebras as logical propositions that can be combined with the operators and, or, not. So I don't quite understand how truth tables are used to characterise elements of finite Boolean algebras. Could that be explained a bit more?
Also, a prominent Stone space is the Cantor set; does it correspond to an interesting Boolean algebra? AxelBoldt 16:06, 13 Feb 2004 (UTC)
Well, to a whole bunch of them. The homeomorphism class of the Cantor set includes many spaces that come up (p-adic integers, typical profinite Galois groups, etc., etc.). Typically the clopen sets are something relatively easy to describe in such examples, and so you get a Boolean algebra that makes some sense. But what makes any one 'presentation' of what is the same Boolean algebra (up to iso) 'interesting'? It's the kind of thing that any two different people might answer two different ways.
Charles Matthews 16:14, 13 Feb 2004 (UTC)
I guess another way of putting roughly the same point is that the self-homeomorphism group of the Cantor set is very large - much bigger than the group of homeomorphisms that preserve the order of reals. The latter is already large, but not impossible to picture. The former is really hard to think about (logicians are welcome to it, in my view). But it clearly respects clopen sets, so gives one a huge way of talking about 'the' Boolean algebra.
Charles Matthews 16:26, 13 Feb 2004 (UTC)
[edit] Move
Why was this article moved from Stone's duality, when that is a term much more likely to be used for the subject of this article (and Stone duality more so)? --- Charles Stewart 19:51, 9 Jun 2005 (UTC)
Wouldn't Stone space be the natural home for this material? Moving it to Stone duality would be like moving group (mathematics) to symmetry (mathematics). Symmetry (mathematics) should simply be a redirect to group (mathematics). --Vaughan Pratt 13:52, 20 August 2007 (UTC)
On second thoughts the current redirect for Symmetry (mathematics), namely to Symmetry in mathematics, is better since group associativity is on the nose whereas that of symmetry in general need not be, e.g. the symmetry of cartesian product, tensor product, etc. --Vaughan Pratt 16:26, 21 August 2007 (UTC)
[edit] Simpler statements of the theorem
Section 1 includes a more traditional and elementary statement of the theorem, which I picked up from Stoll (1963). Those of you with serious training in Boolean algebra should feel free to edit it.
I see that the Wikipedia philosophy community is watching this entry. I have no objections to that fact, but wish to point out that the implications of the Stone theorem for philosophy and logic are less than evident.
This entry should reference a proof that any college senior specializing in algebra should be able to follow givewn close reading. Other than Stoll (1963), I suggest Halmos and Givant (1998: 76-77).132.181.160.42 22:39, 23 September 2007 (UTC)
