[FOM] Infinite closed set absrtracts in Zermelo set theory

[Richard Heck]

On 01/02/2013 05:34 PM, T.Forster at dpmms.cam.ac.uk wrote:
>
> I have been asked the following question by a non-subscriber..
>
>
> The usual formulation of Zermelo has the axiom of infinity in the form 
> that a particular infinite set exists - or an axiom that says that 
> there is a set that contains $\emptyset$ and isclosed under vn Neumann 
> successor, and this easily gives rise to the von Neumann \omega. What 
> happens if we adopt infinity in the form that there is a 
> Dedekind-infinite set? It is known from work of Mathias [Slim models 
> of set theory] that this version of Zermelo does not prove the 
> existence of V_\omega or of the von Neumann \omega. It seems natural 
> to ask if this version proves the existence of any actual named 
> infinite sets *at all* ...i.e., is there a fmla \phi [with only `$x$' 
> free] s.t. this version proves that $\{x:\phi\}$ exists and is 
> infinite? We know from work of Coret that any such \phi would have to 
> be unstratified. Can our readers find such a phi or prove that there 
> is none?'' We conjecture that there is no such \phi.
>
This question is at least in the vicinity of some work done some time 
ago by Gabriel Uzquiano. I do not know if he considered the issue in 
this abstract form, however.

Richard Heck