Does forcing leave some trace in the resulting model?

Question

I'm trying to get initiated in the mysteries of forcing, and I decided to try and do so both by 'studying their patent' and by 'reverse engineering', hoping that each method will be of some help to the other. Here's a very natural question that occurred to me, and one that might either be 'obviously unanswerable' or have an 'obvious answer' for someone more acquainted with the subject. But if this or that is the case, any explanation about why it is so will be appreciated.

Let $V$ be a model for ${\sf ZFC}$ and let $W$ be a submodel of $V$. Is there some criterion to decide if $V$ can be obtained from $W$ by means of a forcing? For instance, how should a given $G\in V$ behave for it to be plausible that $V=W[G]$ for some forcing $\Bbb P\in W$ such that $G\subset\Bbb P$ and $G$ is $\Bbb P$-generic over $W$?

Any descriptions of particular cases, concrete examples, and translations into other branches of mathematics that can provide some intuition are welcome. Also, if there's a topos theoretic version of this question, it would be nice to get in contact with it.

Answer 1

There is a general theorem, due to Laver and Woodin, that identifies in a definable fashion from parameters all inner models $W$ of a model $V$ such that $V$ is a set-generic extension of $W$. See for example

MR2364192 (2009e:03099). Laver, Richard. Certain very large cardinals are not created in small forcing extensions. Ann. Pure Appl. Logic 149 (2007), no. 1-3, 1–6.

This builds on work of Joel Hamkins. The result is essentially that if $\mathbb P$ is a partial order in $W$, $G$ is $\mathbb P$ generic over $W$, $V=W[G]$, and $\delta=(|\mathbb P|^+)^W$, then $W$ is definable in $V$ from the parameter $W_{\delta+1}$.

This gives a uniform way of identifying all possible grounds of $V$, that is, all inner models $W$ such that $V$ is a set-forcing extension of $W$. Hence, we can reason in a first-order way about these models $W$ (the grounds of $V$).

The study of these matters is now called set-theoretic geology, see for instance

MR3304634. Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas. Set-theoretic geology. Ann. Pure Appl. Logic 166 (2015), no. 4, 464–501.

Answer 2

Yes. There is a theorem by Bukovsky which characterises exactly when $W\subseteq V$ is a ground model of $V$ by a $\kappa$-c.c. forcing:

Suppose that for every function $f\in V$ such that $f\colon\alpha\to\mathrm{Ord}$, where $\alpha$ is an ordinal, there is some $g\in W$ such that: (1) $\operatorname{dom}(g)=\alpha$; (2) $|g(\xi)|^W<\kappa$; and (3) $f(\xi)\in g(\xi)$ for all $\xi<\alpha$. Then $W$ is a generic extension of $V$ by a $\kappa$-c.c. forcing.

This is an improvement of a theorem of Vopěnka providing a similar criterion for when $V$ is a generic extension of $W$ by a forcing of size $<\kappa$.

The original proofs are quite old, but Bukovsky recently published a new, and a very nice, proof of these theorems in

Bukovský, Lev, Generic extensions of models of ZFC., Comment.Math.Univ.Carolin. 58,3 (2017) 347–358 ZBL06837070.

This later developed into the fact that the ground model is always definable, and moreover all the ground models are definable by the same formula, where the parameters simply vary over all the sets in the universe. This allows us to study the ground models and their behaviour as a partial order. This field of study emerged about 15 years ago and it is called "set theoretic geology". One good place to start would be

Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas, Set-theoretic geology, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.

And finally, I'd be remiss if I didn't mention that the Axiom of Choice plays a huge role in these results. Very recently Usuba showed that under some conditions we can say a bit more even in $\sf ZF$, but we are still quite a long way from understanding how the behaviour of ground models looks like in $\sf ZF$.