Talk:Stone's representation theorem for Boolean algebras
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)