Is forcing force on us

[Noah Schweber]

>How much do we have to weaken the axioms so that $M[G]$ is no
longer the smallest c.t.m. of the weakened axioms to contain both $M$ and
$G$?

Well, we need to weaken our theory to the point that we cannot build
$\nu[G]$ from a name $\nu\in M$ and the generic $G$ - basically, we need to
get rid of replacement almost entirely. If I understand correctly this
means that even PROVI (see
https://www.dpmms.cam.ac.uk/~ardm/provident_forcing_website.pdf) is too
strong. On the other hand, Zermelo set theory is sufficiently weak (see
section 1 of Mathias' paper "Set forcing over models of Zermelo or Mac
Lane") to have such pathological models/forcings.



 - Noah

On Wed, Aug 26, 2020 at 2:59 PM Timothy Y. Chow <tchow at math.princeton.edu>
wrote:

> I thought of a related question that might be relevant.
>
> It is a standard theorem that if M is a c.t.m. of ZFC, then M[G] is the
> smallest c.t.m. of ZFC containing both M and G (meaning that if N is a
> c.t.m. containing both M and G then N contains M[G]).  But if I understand
> the proof correctly, M[G] is also the smallest c.t.m. of ZF containing
> both M and G, and (I think) the smallest c.t.m. of ZF minus P containing
> both M and G.  How much do we have to weaken the axioms so that M[G] is no
> longer the smallest c.t.m. of the weakened axioms to contain both M and G?
>
> Tim
>
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