[FOM] Cardinality Beyond Regularity and Choice!

[Zuhair Abdul Ghafoor Al-Johar]

But the strong version of Extensionality that I wrote shun Quine atoms as well. I am not really sure if those models you are speaking about can be done with Towers of Singletons (recursive Ur-elements {{..}}) i.e. singletons having every member of their transitive closures as singletons also. i.e singletons that are hereditarily singletons. But I doubt it, since we cannot have more than a countable set of these towers. But in case it does, then we should add the axiom that there exist a unique tower of singletons, in this way we also cut the permutations.

Zuhair

--- On Mon, 12/14/09, Thomas Forster <T.Forster at dpmms.cam.ac.uk> wrote:

> From: Thomas Forster <T.Forster at dpmms.cam.ac.uk>
> Subject: Re: [FOM] Cardinality Beyond Regularity and Choice!
> To: "Zuhair Abdul Ghafoor Al-Johar" <zaljohar at yahoo.com>
> Cc: fom at cs.nyu.edu
> Date: Monday, December 14, 2009, 2:31 PM
> 
> 
> Further to my last: a modifcation rather than an outright
> retraction. Gauntt's model violates extensionality rather
> than foundation beco's it has distinct empty sets
> (urelemente).  What would be needed to answer Zuhair's
> question is a Gauntt model with Quine atoms (objects x =
> {x})
> instead.  I am 99% certain that Gauntt's construction
> works with these objects instead but we live in an imperfect
> world and i should check it - unless some other listmember
> does it first!
> 
>              tf
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