[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
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
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