Categoricity of the complex field in the generic extensions

Question

Let $V[G]$ be a generic extension of $V$ by adding a new Cohen real (or generally a generic extension which adds new reals and do not blow up the power of the continuum). Working in $V[G],$ we can consider the following structures:

1) $(\mathbb{C}^{V[G]}, +, ., 0, 1)$; the field of complex numbers as computed in $V[G].$

2) $(\mathbb{C}^{V}, +, ., 0, 1)$; the field of complex numbers as computed in $V.$

Both of these structures are models of algebraically closed fields of characteristic zero and are of the size of the continuum, so it follows that

$(\mathbb{C}^{V[G]}, +, ., 0, 1) \cong (\mathbb{C}^{V}, +, ., 0, 1)$.

Question. Is it possible to define an explicit isomorphism between the above structures? or is it possible to show that there is no definable isomorphism?

Answer 1

(1) If explicit = definable with parameters from $V$, then the answer is no after adding one Cohen real. For suppose, $\phi(x, y, w)$ ($w \in V$) defines a bijection $x \mapsto y$ from $V \cap \mathbb{C}$ to $V[c] \cap \mathbb{C}$. Then for some $a \in V$ and a Cohen condition $p$, $p \Vdash \phi(a, c, w)$. Let $n$ be outside the support of $p$ and $\pi$ an automorphism of Cohen forcing that acts by flipping the $n$th bit of $c$. Then $\pi(p) = p$ forces $\phi(a, c_n, w)$ where $c_n$ is obtained by flipping the $n$th bit of $c$. Hence $p \Vdash \phi(a, b, w)$ holds for more than one $b$ which is impossible.

(2) If explicit = definable with parameters from $V$ and a real parameter from $V[G]$, then adding $\aleph_1$ Cohen reals will give a negative answer.

Answer 2

In this comment you specified that parameters from the generic extension are fair-game. In this case, the answer is yes, provided the continuum and it's successor have not changed size.

The crux of the idea is that, you can rather naively construct a $\mathbb{P}$-name for an isomorphism using a "respectable" $\mathbb{P}$-name of a well-ordering of $\mathbb{C}^{V[G]}$ with order-type $\vert \mathbb{C} \vert^{V}$.

The naive approach to the construction makes use of the following for the successor step (taking appropriate unions at limit stages.)

Lemma: Given sub-fields $F_0, G_0 \subset \mathbb{C}$, an isomorphism $\varphi_0:F_0 \rightarrow G_0$ and $\xi, \nu \in \mathbb{C}$. If $\xi$ is transcendental over $F_0$ and $\nu$ is transcendental over $G_0$, then there is an isomorphism $\varphi:F_0(\xi)\rightarrow G_0(\nu)$ such that $\varphi\vert_{F_0}= \varphi_0$ and $\varphi(\xi) = \nu$.