[FOM] Extensionality and Church-Oswald constructions

Roger Bishop Jones rbj01 at rbjones.com
Sat Oct 28 04:10:59 EDT 2006

On Friday 27 October 2006 13:18, Thomas Forster wrote:
> (ii)  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)

Is there anything more definite/illuminating which can be said 
about "the right logical kind to be preserved by homomorphisms"?
(i.e. which might make help me back up or dispell the hunch that 
my closure principle is of this kind).

Since mailing my question to FOM (it took 10 days to arrive!) I 
have had time to realise that the constructions in your book 
would probably all fail if they were not extensional in the 
first instance.  The most obvious reason is that they all (so 
far as I recall) have complementation as an operation and if 
complementation is defined in a non-extensional domain it will 
not survive taking quotients to realise extensionality.

Though my construction is similar in starting out with a well- 
founded model and filling it out, it is dissimilar in its 
conception of what the non-well-founded sets should be.
It is based on a narrower conception of polymorphic function and 
predicate than that realised through stratified abstraction.
This conception is not closed under complementation.

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

Thanks, I will.

Roger Jones

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