# [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

tf

www.dpmms.cam.ac.uk/~tf

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