I am a college student with very little mathematics background (up to Calculus 152 at Rutgers University), but have become increasingly interested in computing and mathematics in the last year. I am currently reading Godel, Escher, Bach, which offers a lightweight, non-rigorous exposition of topics in logic and number theory. I don't even know if my question makes sense :) so please excuse me if this comes out as gibberish:
What is the relationship between Algebraic structures like Fields and Rings, and Formal systems? I'm pretty sure Peano arithmetic is a formal system, and wonder how fields and rings apply to or within Peano. Are the arithmetic "operations" applied to fields and rings formal systems embedded within the given field or ring? Or are fields and rings specific kinds of formal systems (i.e. every field is a formal system, but not every formal system is a field)?
Again, this question might be relying on a radically misguided understanding of the terms I am using, so it may not even make sense. If my answer makes no sense, maybe you can lead me to some nice introductory literature. I "kind of" understand what fields and rings are, but I understand much better what a formal system is, and I understand Godel numbering is, if that helps.