genField_countable

In the Primitive Recognition Calculus, the generable reals relative to a countable family of named constants $\kappa$ are defined as the subfield of $\mathbb{R}$ obtained by closing the range of $\kappa$ under the field operations. Equivalently, they are everything obtainable from the rationals and the named constants by finitely many additions, multiplications, and inverses.

The load-bearing upstream fact is that any subfield of $\mathbb{R}$ generated by a countable set is countable: adjoining countably many reals (algebraic or transcendental) to $\mathbb{Q}$ never escapes countability. That is the field-level analogue of the algebraic-closure countability lemma for Delta-reals. The range of a sequence $\kappa:\mathbb{N}\to\mathbb{R}$ is automatically countable, so the generable carrier inherits countability from that general closure theorem.

One-line term proof. Apply the general lemma that subfield closure of a countable set of reals is countable, feeding it the standard fact that the range of any sequence $\kappa:\mathbb{N}\to\mathbb{R}$ is a countable set. No further case analysis or cardinal arithmetic is performed at this site.

This is the countability half of the generable-carrier package. Downstream, it is the sole ingredient that makes the generable field a proper subset of $\mathbb{R}$: if the carrier equalled the continuum it would be uncountable, contradicting this theorem. The objecthood registry packages the pair (countable and proper) as the classification of every countable constant inventory as a permitted operational carrier. Together with the operational-carrier closure properties (contains $\mathbb{Q}$ and the named constants, closed under $+$, $\cdot$, and negation), it underwrites the Phase-2 headline that display can exceed generation: there exist reals realized by Delta-real protocols that lie outside every countable generable subfield.