The Newman Objection to Ramsey-Sentence Structuralism, II

I posted on this topic a couple of months ago. This is a quick follow up. The Newman Objection to Ramsey-sentence (= Carnapian) structuralism can be given quickly, if we first use two definitions and a lemma. Below, $L$ is an interpreted language with a well-defined O/T distinction; $\Theta$ in $L$ is a finitely axiomatized theory. $\Re(\Theta)$ is then the Ramsey-sentence of $\Theta$. (The details here can become very, very messy; but it simply muddies the water to go into all that.)

Definition 1. Ramsey-sentence structuralism (Carnap)
The synthetic content of $\Theta$ = $\Re(\Theta)$.

Definition 2. Constructive empiricism (van Fraassen)
The content of a theory $\Theta$ that we may accept is: $\Theta$ is empirically adequate.

Ramsey-sentence Lemma
$\Re(\Theta)$ is true iff $\Theta$ has an empirically correct model $\mathcal{M}$ satisfying a cardinality condition on the entities it thinks are unobservables.

Then the objection can be put like this:

Newman Objection
Ramsey-sentence structuralism $\approx$ Constructive empiricism

The argument for this is: let us suppose that Ramsey-sentence structuralism is true and Constructive empiricsm is true. Then, because of the Lemma, it follows that accepting $\Re(\Theta)$ $\approx$ accepting $\Theta$'s empirical adequacy (that is, the sole difference is the cardinality commitment). QED.

Like all philosophical arguments, the technical regimentation is not 100% tight, and many possible queries can arise (in my view, these are: the precise scheme of formalization for theories; the precise definition of "empirically adequate"; and the range of the second-order quantifiers). Even so, under various clarifications, precisifications, modifications of definitions, etc., the conclusion remains more or less invariant.