**Definition 1:** A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has an "initial member" $M$ if each member of $\mathcal{K}$ is a forcing extension of $M$ for some partial order $\mathbb{P}\in M$ and some $\mathbb{P}$-generic $G$ over $M$. 


**Definition 2:** An extension $T$ of $\text{ZF}$ has an "initial c.t.m" if the collection of all countable transitive models of $T$ has an "initial member". 


**Question 1:** Assuming some consistency assumptions is it consistent that $\text{ZF}$ has an initial c.t.m?


**Question 2:** Assuming some consistency assumptions is it consistent that $\text{ZF}$ has a consistent extension like $T$ with an initial c.t.m? 



Does ZF have an initial model?
Your question has a certain affinity with the concept of a *solid bedrock* model, which arises in the theory of [set-theoretic geology](http://jdh.hamkins.org/set-theoreticgeology/). Namely, $W$ is *bedrock* for $V$ if $V$ is a forcing extension of $W$ and $W$ satisfies the [ground axiom](http://en.wikipedia.org/wiki/Ground_axiom), meaning that it is not a forcing extension of any deeper ground, or in other words, that it is minimal among all the grounds of $V$. The model $W$ is a *solid bedrock* if $W$ is least among all the grounds of $V$. 


Although assertions about whether there is a bedrock or whether there are grounds of a certain nature seem at first to be second-order assertions about $V$, because they quantify over the inner models $W$ that might be grounds, in fact these are all first- order expressible in the language of set theory. The reason is that the collection of grounds of $V$ is uniformly definable, in that there is a definable family of classes $W\_r$ such that every $W\_r$ is a ground of $V$; every ground of $V$ is $W\_r$ for some $r$; and the relation $x\in W\_r$ is definable in $x$ and $r$. Thus, one may quantify over the collection of grounds by quantifying over the parameter $r$ used in this definition.


In his dissertation, Jonas Reitz proved that there are *bottomless* models $V$, which have no bedrock models; that is, a bottomless model $V$ can be realized as a set-forcing extension $V=W[G]$ of a ground $W$, but one can always go deeper, and realize $W=W\_0[G\_0]$ as a forcing extension of a still deeper ground, with no bottom. 


