I am an undergrad working on my senior thesis. I am trying to come up with a first order axiomatization for the ring of period numbers. However, due to the nature of the ring I am trying to describe, I keep finding it necessary to quantify over a specific family of function symbols. Alternatively, I might be able to make it work by quantifying over the rationals as well as a subset of my broader structure.
My gut tells me that using a 2-sorted logic would make it more difficult to use the model theory tools I've learned to reason about these numbers, and it feels much messier. That being said, I am fairly new to the topic. My only real experience with model theory is half of Marker's text, and he does not go into much detail about many sorted logics, taking only two pages to briefly define them. It might be possible my desire to avoid 2-sorted logic is unfounded. I would appreciate any guidance on how to proceed, or any suggested reading.