Re: Prime Ideal Theorem implies Excluded Middle?

On Sat, 22 Mar 2003, C.F.Townsend wrote:

Dear all, does anyone know a reference for the question:

Prime Ideal Theorem implies the Excluded Middle ?

My paper "Another condition equivalent to De Morgan's Law" (Commun Alg. 7 (1979), 1309-1312) shows that the statement "Every maximal ideal is prime" for distributive lattices (or for Boolean algebras) is equivalent to De Morgan's law. In a localic Set-topos (assuming AC in Set) the Maximal Ideal Theorem holds; hence in the topos of sheaves on an extremally disconnected locale the Prime Ideal Theorem holds (cf. Elephant, D4.6.15). But such a topos needn't be Boolean. Peter Johnstone