# [FOM] Infinite closed set absrtracts in Zermelo set theory

Richard Heck richard_heck at brown.edu
Thu Jan 3 15:17:11 EST 2013

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

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