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

[Roger Bishop Jones]

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 
discourse.

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