Terminology "totally transcendental" in model theory

Question

The following is a definition of totally transcendental (a concept in model theory):

Where does the terminology "totally transcendental" come from? Does it have any connection with other notions of transcendence in mathematics (e.g. transcendental numbers)?

Answer 1

The name comes from the fact that Morley called his rank "transcendental rank". And yes, it is directly related to the notion of transcendental number/element in field theory. Quoting from Morley's original paper Categoricity in Power:

$p\in S(A)$ is algebraic if $p\in \mathrm{Tr}^0(A)$; $p$ is transcendental in rank $\alpha$ if $p\in \mathrm{Tr}^{\alpha}(A)$.

Here $S(A)$ is the type space in one variable over $A$, and $\mathrm{Tr}^\alpha(A)$ is the set of types of Morley rank $\alpha$ (though Morley's original definition looks rather different from the modern definitions).

In a footnote to this line, Morley writes:

The terminology algebraic and transcendental are suggested by the theory of algebraically closed fields of characteristic $0$, see Example 1 below.

In Example 1, Morley carries out an analysis of the type space $S(A)$ in one variable over a subset $A$ of an model of $\mathrm{ACF}_0$. He writes $\Delta(A)$ for the subfield generated by $A$. Quoting again:

Thus $S(A)$ consists of: (1) isolated points corresponding to the distinct elements of $\Delta(A)$ and to the algebraic extensions of $\Delta(A)$, and (2) a single limit point corresponding to the transcendental extensions of $\Delta(A)$.

Finally, Morley defines:

We say $T$ is totally transcendental if $S^{\alpha_T}(A) = \emptyset$ for some (and hence every) $A\in \mathcal{N}(T)$.

Unpacking the notation here, this is equivalent to saying that every type in every $S(A)$ is in $\mathrm{Tr}^\alpha(A)$ for some ordinal $\alpha$, i.e. every type has ordinal-valued Morley rank.

So it seems Morley viewed types of ordinal-valued rank $>0$ as generalizing the transcendental type in $\mathrm{ACF}_0$. From this point of view, it makes sense to call a theory totally transcendental if all of its types have rank (i.e. are algebraic or transcendental, possibly of higher rank). He may also have been thinking of the fact that the rank of a type $p(x_1,\dots,x_n)$ in multiple variables over a field $K$ is equal to the transcendence degree of $K(a_1,\dots,a_n)$ over $K$, when $(a_1,\dots,a_n)$ is a realization of the type. But Morley only considered type spaces in a single variable in this paper.

The definition of totally transcendental you quote in your question is just an equivalent to the definition in terms of rank, which is sometimes easier to use, check, or refute.