Indeed, there's a way to get a semantics for free, based on the syntax alone. For example, in the first order logic this is the Herbrand interpretation

https://en.wikipedia.org/wiki/Herbrand_interpretation

The point of mathematical semantics is that for any given theory, we can have other interpretations that don't just interpret symbols as themselves.

So we could conceivably imagine an interpretation where ∞ doesn't just mean literally ∞ and nothing more.