Re: Complete Heyting algebras

Re: Complete Heyting algebras

[JeanBenabou]

Of course not !
Elementary toposes are models of a first order theory (this is also  
true for elementary toposes with an NNO) . Thus there are countable  
models. In such a model the answer is no.

Le 20 nov. 07 à 13:10, Lutz Schroeder a écrit :

> Is the Heyting algebra of global elements of the classifier in an
> elementary topos always complete? (Of course, the classifier is  
> complete
> internally; here, I mean externally complete.) I suspect not, but I
> can't presently think of a natural counterexample.
>
> Thanks,
>
> Lutz Schröder
>
>
> -- 
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Re: Complete Heyting algebras

[Prof. Peter Johnstone]

On Tue, 20 Nov 2007, Lutz Schroeder wrote:

> Is the Heyting algebra of global elements of the classifier in an
> elementary topos always complete?

The answer is no: any Boolean algebra, complete or not, can occur
as Sub(1) in a topos (see Exercise 9.11 in my old Topos Theory book),
and any quotient of a complete Heyting algebra (by a finitary Heyting
congruence -- such quotients needn't be complete) can occur, by use of
the filterpower construction (cf. my paper with Murray Adelman "Serre
classes for toposes", Bull.Austral.Math.Soc. 25 (1982), 103-115).
There are examples due to Peter Freyd
of Heyting algebras which can't occur as Sub(1) in a topos
generated by subobjects of 1. For a long time it was an open
problem whether any Heyting algebra can occur as Sub(1) in a topos
(without the restriction on generators): Dito Pataraia has recently
announced a positive solution to this problem. I have heard a seminar
talk about his solution, and seen half of a preprint, but haven't yet
managed to understand the other half of his construction.

Peter Johnstone

Complete Heyting algebras

[Lutz Schroeder]

Is the Heyting algebra of global elements of the classifier in an
elementary topos always complete? (Of course, the classifier is complete
internally; here, I mean externally complete.) I suspect not, but I
can't presently think of a natural counterexample.

Thanks,

Lutz Schröder


-- 
------------------------------------------------------------------
PD Dr. Lutz Schröder                  office @ Universität Bremen:
Senior Researcher                     Cartesium 2.051
Safe and Secure Cognitive Systems     Enrique-Schmidt-Str. 5
DFKI-Lab Bremen                       FB3 Mathematik - Informatik
Robert-Hooke-Str. 5                   Universität Bremen
D-28359 Bremen                        P.O. Box 330 440
                                      D-28334 Bremen
phone: (+49) 421-218-64216            Fax:   (+49) 421-218-9864216
mail: Lutz.Schroeder@dfki,de
www.dfki.de/sks/staff/lschrode
------------------------------------------------------------------


-------------------------------------------------------------
Deutsches Forschungszentrum für Künstliche Intelligenz GmbH
Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern

Geschäftsführung:
Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender)
Dr. Walter Olthoff

Vorsitzender des Aufsichtsrats:
Prof. Dr. h.c. Hans A. Aukes

Amtsgericht Kaiserslautern, HRB 2313
-------------------------------------------------------------