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.