[FOM] Infinite closed set absrtracts in Zermelo set theory

T.Forster at dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Wed Jan 2 17:34:41 EST 2013

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.

           Happy New Year



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