You may be thinking of the 'downward' one of the Lowenheim-Skolem theorems. An axiomatization cannot require a model that is strictly larger than the set of sentences generated by the language. Most importantly, every countable axiom set has a model with only countably many elements.
So in particular, any countable axiomatization of the Real numbers has a model with countably many elements. But the real numbers as we use them are both uncountable and well-defined, they are just not the smallest equivalent model to whatever view we take of the Reals at any given point in time.
But unless you drag infinite cardinals into it, this seems directly contradicted by the notion of language. Finite human languages certainly define a world that includes a lot more than the language, if only to a limited degree. Theoretically, every language is countably infinite in its number of sentences. But the actual sentences that can be meaningfully understood, given the limitations of human intelligence are a finite subset of the physical things. The number that will ever be said is even smaller.
And this is true of specific mathematical languages. Clearly the finite axiom set of ZFC, which is finite, defines an infinite world of sets. The finite algebraic model of the integers defines an infinite sequence. And so on.
So as a principle, this may be true in some situations. But even then, it is demonstrably not true of some finite things.