What is the dimension of the mathematical universe?

My co-authors and I introduced a notion of dimension for forcing extensions in the following paper:

• Hamkins, Joel David; Leibman, George; Löwe, Benedikt, Structural connections between a forcing class and its modal logic, Isr. J. Math. 207, Part 2, 617-651 (2015). ZBL1367.03095, arxiv:1207.5841, blog post.

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. $\Box$

Let me first give the motivations of the definitions given in my paper.

So suppose $F \subseteq K$ be fields. Note that $[K:F],$ the dimension of $K$ over $F$ considered as a vector space is equal to $\kappa$ iff $\kappa$ is maximal length of a chain $F=F_0 \subset F_1 \dots F_\kappa=K$, where each $F_i$ is a vector space over $F$. My definition of dimension is essentially taken from this fact.

The note is that based on Hamkins definition, we don't have finite dimension, while in my case for example a one dimenional extension of a model of ZFC is just a minimal generic extension, which in my opinion seems reasonable.

Also note that if $[K:F]=\kappa$  and if $\lambda < \kappa,$ then there exists a vector space $G$ over $F$ with $F \subset G \subset K$ and $[G:F]=\lambda.$ However this is not true in general nor for the Hamkins definition nor for my definition.

Also my definition allows to consider pairs $V \subset W$, where $W$ is not necessarily a set forcing extension of $V$, and for this reason I have defined the upward generic dimension and the downward generic dimension, the latter being related to set theoretic geology.

Another important notion is the notion of independence.Note that $x, y \in K$ are independent over $F$ iff $F[x] \cap F[y]=F$ (where for $X \subseteq K,$ $F[X]$ is the vector space generated by $F \cup X$). Note that a subset $X$ of $K$ is independent over $F$ iff for any finite disjoint subsets $X_1, X_2$ of $X$ we have $F[X_1] \cap F[X_2]=F$. On the other hand, using a theorem of Solovay, it is natural  for $V \subset W$ and for $x, y \in W$ to say that $x$ and $y$ are independet over $V$ if they are mutually generic. This is  my motivation for $\aleph_0$-transendence degree between models of set theory, of course in this case we can define $\kappa$-transendence degree for any $\kappa \geq \aleph_0.$  

Note that as it is stated in the comments, in general we don't need for example $x$ and $y$ to be mutually generic over a model $V$ to have  $V[x] \cap V[y]=V$, however the assumption of they being mutually generic has some advantages, for example we can form $V[x, y]$ and it is again a model of ZFC, while in general this may not be the case.

Regarding your update, I don't see any essential similarities between these two concepts:

1) Suppose $\mathbb{P}$ is a forcing notion of size $\kappa$ which gives a minimal generic extension of the universe and let $\mathbb{Q}=Add(\omega, \kappa).$ Then:

(1-a) In the sense of Hamkins definition $[V^{\mathbb{P}}, V]=[V^{\mathbb{Q}}, V]=\kappa$.

(1-b) In the sense of my definition $[V^{\mathbb{P}}, V]=1$ while $[V^{\mathbb{Q}}, V]=\kappa^+$.

2) Suppose $\mathbb{P}$ and $\mathbb{Q}$ gives minimal generic extensions of the universe and are of size $\kappa$ and $\lambda$ respectively where $\kappa < \lambda$. Then

(2-a) In the sense of Hamkins definition $[V^{\mathbb{P}}, V]=\kappa$ and $[V^{\mathbb{Q}}, V]=\lambda$.

(2-b) In the sense of my definition $[V^{\mathbb{P}}, V]=[V^{\mathbb{Q}}, V]=1$.