[FOM] References on continuum hypothesis in non-well-founded set theory

Roger Bishop Jones rbj at rbjones.com
Fri Oct 24 03:44:46 EDT 2008

On Thursday 23 October 2008 10:29:00 Andrej Bauer wrote:
> I have been contacted by an amateur who came up with what he claims to
> be a model of non-well-founded set theory in which the continuum
> hypothesis is false.
> As I am not an expert in this field, I would very much appreciate some
> references about this subject. His particular model seems _not_ to
> satisfy replacement, foundation and separation. This means that it would
> be useful to have references to works discussing what happens with
> continuum hypothesis when various standard axioms are missing (not just
> foundation).

There are various ways in which non-well-founded interpretations of set theory 
can be constructed from well-founded interpretations yielding a 
non-well-founded interpretation of which the well-founded part is isomorphic 
to the original.

Hence, by starting with a well-founded interpretation of set theory in which 
the continuum hypothesis is false one can construct a non-well-founded 
interpretation with the same characteristic.

Even if your original interpretation was a model for ZFC, you would not expect 
the axioms of ZFC to hold without qualification in the non-well-founded model 
constructed from it.  They hold in the well-founded part, and some of them 
would have to be reformulated to avoid asserting them of the whole domain of 

I'm not well acquainted with the literature but there is a description of some 
ways of doing this in Thomas Forster's book "Set Theory with a Universal Set" 
(under the heading "Church Oswald Models") and there are papers in his 
bibliography on the methods, see. http://www.dpmms.cam.ac.uk/~tf

Roger Jones

More information about the FOM mailing list