Re: when are Lindenbaum-Tarski algebras complete?

[Dana Scott <dana.scott@cs.cmu.edu>]

On Nov 21, 2007, at 6:33 AM, Thomas Streicher wrote:

> I also suspect that Sub(1) of the free topos with nno is not complete.
> But countability does not suffice for refuting completeness (the
> ordinal
> \omega + 1 is an infinite countable cHa which nevertheless is
> complete).
>>
> From Goedel's Theorem for HAH (higher order intuit. arithmetic) it
> follows
> that Sub(1) of the free topos with nno is not atomic. But that also
> doesn't
> suffice for refuting completeness.
>
> On p.169 of Freyd, Friedman and Scedrov's paper
> "Lindenbaum algebras of intuitionistic theories and free
> categories" (APAL 35)
> they claim "Lindenbaum algebras are almost never complete" but don't
> give
> a proof.


Ah, but if a Heyting algebra is complete, then so is the Boolean algebra
of all not-not-stable elements.  Familiar example: the regular open
subsets of a topological space form a complete Boolean algebra.

As remarked, the Sub(1) of the free topos with nno is not atomic, and with
reference again to Godel's theorem via the not-not translation, the
Boolean algebra of not-not-stable elements is also non atomic.  But all
countable, non-atomic Boolean algebras are isomorphic to the clopen
subsets of the Cantor space (or the Lindenbaum algebra of classical
propositional calculus, or the free Boolean algebra on countably many
generators).  That algebra is not complete -- as can be seen in many ways.
Q.E.D.