A Boolean model of ultrafilters

✍ Scribed by Thierry Coquand

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 557 KB
• Volume: 99
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(99)00006-8

✦ Synopsis

We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let B" denote the algebra of sequences (x,,), x,, E B. Let us write PA E B1 the sequence such that pi(i) = 1 if i<k and pi(i) = 0 if k < i. If x E B, denote by x* E B,' the constant sequence x* = (I,.Y,x,. .). We define a Boolean meamre algebra to be a Boolean algebra B with an operation p: B' -B such that p(p~ ) = 0 and ,c(x*) = x. Any Boolean measure algebra can be used to model non-principal ultrafilters in a suitable sense. Also, we can build effectively the initial Boolean measure algebra. This construction is related to the closed open Ramsey Theorem (J. Symbolic Logic 38 ( 1973 ) 193-198.