[FOM] Extensionality and Church-Oswald constructions

Thomas Forster T.Forster at dpmms.cam.ac.uk
Fri Oct 27 08:18:04 EDT 2006

My initial reaction is twofold

(i)  It's best to nip the problem in the bud.

(ii) (this is the more important consideration, i think).  If you
do not enforce extensionality from the start you have the problem of 
showing that the desirable properties of the construction are 
preserved in the quotient.   Since the sentences that axiomatise set 
theory are not of the right logical kind to be preserved by homomorphisms, 
one has to put in a lot of extra work, which leads us back to (i)

Incidentally, don't look at the chapter of my book, read instead the 
updated and improved version of it available from my home page.

On Tue, 17 Oct 2006, Roger Bishop Jones 

> 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).
> Roger Jones
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