>>13061104Assume the original, finite theory that we infinitely expanded in
>>13056699 was just the axioms of Peano Arithmetic (save the second-order induction axiom, and some of the redundant identity axioms) formulated in a first-order language.
Any structure that satisfies the infinitely expanded theory *also* satisfies the original, finite theory, obviously.
Which means your nice little version of Peano Arithmetic done with first-order logic, besides having well-behaved models that look like you intended, also have unintended, even bizarre models like the one I described, with some numbers infinitely distanced from others, whether you like it or not, because of Compactness.
And worse, while the model I presented at least has a countable domain (effectively the union of two countably infinite sets--which is countable), just like your intended model, in general, once you have a theory that axiomatizes models with inifinite domains, you lose the ability to constrain the cardinality of *all* models (if it has any) that satisfy it, thanks to the Lowenheim-Skolem theorems (in this case, Tarski's upward variant).
That means your basic theory and the expanded version, if they have any models at all, also have models with *uncountably infinite* domains--contrary to our intentions.
And likewise, a theory that axiomatizes arithmetic on real numbers (which are uncountable), or even the axioms of some system of set theory (e.g., ZFC) formulated in a first-order language have models with only *countably infinite* domains.
This is a well-known shortcoming of first-order logic. You can express concepts like "all objects are finitely distanced from each other" and even ensure all models that satisfy your theory have exactly the same cardinality (e.g., 42)--but only if the domains are all finite.
Once your theory axiomatizes models with *infinite* domains, you lose this specificity.
