[FOM] Extensionality and Church-Oswald constructions
Roger Bishop Jones
rbj01 at rbjones.com
Tue Oct 17 02:08:54 EDT 2006
I am at present engaged in the construction of a model for a set
theory with a universal set (not NF or NFU).
The method I am using is similar in some respects to the method
described in Chapter 4 of Forster's book on set theory with a
universal set, and called there "Church-Oswald constructions".
This involves taking a model of a well-founded set theory and
extending it to achieve closure under additional operations
which yield non-well-founded sets.
In the examples shown by Forster care is taken to use
constructions which yield extensional results.
I however, had in mind using a construction which will not give
an extensional relationship and then obtaining from this an
extensional relation over equivalence classes of elements from
the domain of the constructed model using the smallest
equivalence relation which will give an extensional result.
I am loth to make the construction considerably more complex for
the sake of extensionality (which I think it would have to be in
my case) if I can easily fix the problem later.
Does anyone know reasons why getting extensionality in the
initial construction might be necessary or desirable?
(presumably problems with the obvious method of subsequently
trading up to an extensional relationship).
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