Prime Ideal Theorem implies Excluded Middle?

Prime Ideal Theorem implies Excluded Middle?

[C.F.Townsend]

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

Prime Ideal Theorem implies the Excluded Middle ?

Certainly, Axiom of Choice implies Excluded Middle, but I have convinced
myself that the weaker statement is true and would be grateful for any
pointers.

Regards, Christopher Townsend

PS there are definitely formulations of the PIT that do not use negation.
E.g. it is equivalent to the statement that for every Boolean alg. B if x in
B has the property that f(x)=0 for every Boolean alg. homomorphism
f:B->Omega, then x=0.

Re: Prime Ideal Theorem implies Excluded Middle?

[Prof. Peter Johnstone]

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