The reals can be studied as models of many theories. They (with the operation +, relation = and element 0) are a model of the axioms of an abelian group. They are also a model of the axioms of a group. The reals with (+, , 0, 1, =) are a model of the axioms of a field. The reals with (+, , 0, 1, =, <) are a model of the axioms of an ordered field. Etcetera…
Models are things. Theories are collections of statements about things. A model can satisfy many theories; a theory can have many models. I agree completely with So8res statement that it is important to keep the two straight.
In addition, your example of Dedekind completeness is an awkward one because the Dedekind completeness axiom is a good example of the kind of thing you can’t say in first order logic. (There are partial ways around this, but I’m trying to keep my replies on the introductory level of this post.) But I can just imagine that you had distinguished the reals and the rationals by saying that, in R, ∃ x : x^2=1+1 is true and in Q it is false, so I don’t need to focus on that.
The reals can be studied as models of many theories. They (with the operation +, relation = and element 0) are a model of the axioms of an abelian group. They are also a model of the axioms of a group. The reals with (+, , 0, 1, =) are a model of the axioms of a field. The reals with (+, , 0, 1, =, <) are a model of the axioms of an ordered field. Etcetera…
Models are things. Theories are collections of statements about things. A model can satisfy many theories; a theory can have many models. I agree completely with So8res statement that it is important to keep the two straight.
In addition, your example of Dedekind completeness is an awkward one because the Dedekind completeness axiom is a good example of the kind of thing you can’t say in first order logic. (There are partial ways around this, but I’m trying to keep my replies on the introductory level of this post.) But I can just imagine that you had distinguished the reals and the rationals by saying that, in R, ∃ x : x^2=1+1 is true and in Q it is false, so I don’t need to focus on that.