Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the existing ones in a fairly controlled and comprehensible way and connect them to one another through forcing extensions.
Since the very beginning, the true nature of this rather bizarre construction method has been a matter of debate. A widely accepted (?) analogy emphasizes on a degree of similarity between forcing method in set theory and field extensions in Galois theory where one constructs new fields using the existing ones by adding new elements. Besides some papers and textbooks, a good description of this analogy could be found in this MathOverflow good oldie: Forcing as a new chapter of Galois Theory?.
However, as Dorais mentioned in his answer, despite the undeniable similarity, this analogy is not as perfect as what it seems at the first glance. So, forcing is not a one hundred percent field extension-like construction anyway. This fact causes some confusion concerning the possible translation of Galois theory machinery into the language of forcing and set theory. It is often not immediately clear what the forcing analogy of a given notion in the field extensions would be. Also, sometimes there is more than one approach towards defining a corresponding Galois theoretic notion in forcing so that one needs to decide which the natural one is.
One such central concept in Galois theory is the notion of the degree of a field extension, that is the dimension of the extended field while viewed as a vector space over the ground field. The question that arises here is whether there is any corresponding well-defined, well-behaved and natural similar notion in set-theoretic forcing.
One may think if a forcing extension could be viewed as a kind of vector space-like structure over the ground model. In this sense, there might be a kind of basis associated with any such pair of models which itself may satisfy some uniqueness properties that give rise to the existence of a well-defined notion of (relative) dimension of a generic forcing extension with respect to its ground model. Our axiomatic expectations of the possible behavior of any such notion of basis/dimension is also an interesting topic to explore even before defining any such notion. Of course, we need them to behave as natural as possible and resemble their Galois theoretic counterparts fairly closely.
Question. What are examples of defining a notion of dimension or basis for forcing generic extensions in the set-theoretic literature (possibly close to the same fashion that exists for the field extensions and vector spaces)?
I couldn't find much along these lines in the literature except a short unpublished note of Golshani in which he takes a Galois theoretic approach towards dimension in forcing by dealing with mutually generic sequences and the chain of forcing extensions. It also shares many features with Hamkins' set-theoretic geology project. However, this notion of forcing dimension seems not to be complete enough to cover all aspects of a vector space view towards generic extensions but could be a really good starting point anyway.
Remark. As Peter stated in his comment, absoluteness would be an issue while defining a notion of (relative) dimension in the set-theoretic multiverse. One may ask, from whose perspective are you trying to calculate the relative dimension of two set-theoretic universes and why? In fact, due to the highly contradictory views of different models of $\sf ZFC$ towards anything beyond the narrow scope of universally absolute properties, there is not more than a little hope to obtain an absolute notion of forcing dimension after all. However, one may argue that absoluteness is not a crucial condition in the long list of our expectations from a possible natural definition of dimension for generic models. The same situation happens in the Hamkins' Well-foundedness Mirage Principle stating that "Every universe $V$ is non-well-founded from the perspective of another universe" and so there is no standard model of $\sf ZFC$.
Update. According to the Joel and Mohammad's answers, it turned out that there is more than one approach towards developing a dimension theory for forcing extensions; each with their own characteristics. While Joel's definition requires all forcing dimensions to be infinite, Mohammad's approach allows finite dimensions as well as infinite ones. Also, as Monroe pointed out both definitions fail to satisfy the so-called downwards closure property.
In general, Joel's approach sounds a little bit vector space-like to me while Mohammad's reminds me the Krull's dimension in commutative algebra. There is also a chance that these different dimensions be related or coincide with each other under certain circumstances.
The point is that in the absence of a general common sense about the expected behavior of a nice forcing dimension, it is not immediately clear whether the mentioned features are flaws or advantages of the presented definitions. Maybe one needs to fix some abstract list of required properties for any such dimension operator and then search for its existence in the forcing extensions. Anyway, if there is any suggestion for such an axiomatic approach towards forcing dimension, I will be so happy to hear about it.
Best Answer
My co-authors and I introduced a notion of dimension for forcing extensions in the following paper:
Specifically, for any forcing extension $V\subset V[G]$, we defined the essential size of the extension to be the smallest cardinality in $V$ of a complete Boolean algebra $\mathbb{B}\in V$ such that $V[G]$ is realized as a forcing extension of $V$ using $\mathbb{B}$, so that $V[G]=V[H]$ for some $V$-generic $H\subset\mathbb{B}$. More recently, I have been inclined to call this the forcing dimension of $V[G]$ over $V$.
This can indeed be seen as a dimension, in light of the following (essentially lemma 23 of the paper above):
Theorem. If $V\subset V[G]$ has essential size $\delta$, then the essential size of any further extension $V[G][H]$ over $V$ is at least $\delta$.
Proof. By combining the forcing into an iteration, we may view $V[G][H]$ as a single-step forcing extension of $V$, and so it has some essential size over $V$. Since we have an intermediate model $V\subset V[G]\subset V[G][H]$, it follows by the intermediate model theorem that $V[G]$ can be realized as an extension by a complete subalgebra of that forcing notion. So the smallest size of a complete Boolean algebra realizing $V[G]$ is not larger than the smallest size of a compete Boolean algebra realizing $V[G][H]$ over $V$. $\Box$
We had used the fact that there is a definable dimension, in the forcing extensions over $L$, to show in general circumstances that the modal logic of $\Gamma$-forcing over $L$ is contained in S4.3, for a wide collection of forcing classes $\Gamma$.
Monroe Eskew points out in the comments that, contrary to my initial thoughts about this, the size of the smallest partial order giving rise to the extension will also serve as a forcing dimension. The reason is that the density of a complete subalgebra of a complete Boolean algebra is at most the density of the whole algebra, simply by projecting any dense set of the larger algebra to the subalgebra. It follows by the same argument as in the theorem above that the poset-based forcing dimension of any intermediate model in a forcing extension is bounded by the minimal size of a partial order giving rise to the whole extension.
I propose that we officially adopt the poset-based notion as the forcing dimension of a forcing extension $V\subset V[G]$, denoting this dimension by $\left[V[G]\mathrel{:}\strut V\right]$. We may now observe the following attractive identity, confirming the suggestion of Will Brian.
Theorem. For any successive forcing extensions $V\subset V[G]\subset V[G][H]$, we have $$\left[V[G][H]\mathrel{:}\strut V\right]=\left[V[G]\mathrel{:}\strut V\right]\cdot\left[V[G][H]\mathrel{:}\strut V[G]\right].$$
Proof. Suppose that $G\subset\mathbb{P}\in V$ and $H\subset\mathbb{Q}\in V[G]$, where these are the minimal-size partial orders realizing the extensions. Suppose that $\mathbb{Q}$ has size $\kappa$ in $V[G]$. So without loss there is $\mathbb{P}$-name for a relation $\dot\leq$ on $\kappa$, such that $\mathbb{Q}=\langle\kappa,{\dot\leq}_G\rangle$ in $V[G]$. We can now use the partial order $\{(p,\check\alpha)\mid p\in\mathbb{P},\alpha<\kappa\}$, which is dense inside $\mathbb{P}*\dot{\mathbb{Q}}$, to realize $V\subset V[G][H]$. This shows $\leq$ of the desired identity.
Conversely, if we can realize $V[G][H]$ as a forcing extension of $V$ by some partial order $\mathbb{R}$, then $V[G]$ arises as a subforcing notion, and $V[G][H]\supset V[G]$ arises as quotient forcing. The quotient forcing $\mathbb{R}/G$ can be thought of as the conditions in $\mathbb{R}$ that are compatible with every element of (the image of) $G$ in (the Boolean completion of) $\mathbb{R}$. So the smallest partial order giving rise to $V[G][H]$ over $V[G]$ is at most the size of the smallest partial order giving rise to $V[G][H]$ over $V$, as desired. $\Box$