# [FOM] Why would one prefer ZFC to ZC?

T.Forster@dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Sat Jan 30 02:24:43 EST 2010

You can take the axiom of infinity in at least three forms

(i)   The Von Neumann omega exists
(ii)  Thee is a Dedekind-infinite set
(iii) V_\omega exists

In ZF they are all equivalent. Without replacement they are all
inequivalent. See e.g. Adrian Mathias' *slim models* paper in the JSL.

tf

On Jan 29 2010, Bill Taylor wrote:

> -> There is nice a series of papers by Gabriel Uzquiano, deriving from
> his -> dissertation, that show that what one might have thought were
> equivalent -> forms of the axiom of infinity come sharply apart in ZC.
> That has always -> struck me as more of an "internal" reason to be
> dissatisfied with ZC. -> -> Richard Heck
>
> That is a very interesting observation! Could you please give us an
> example case of this phenomenon, so that we can get an idea of the
> concepts involved?
>
>It would be a kindness.
>
>Bill Taylor
>
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