Are algebraic structures first order theories?

Question

When we talk about algebraic structures, we can say things like: ($\mathbb{Z}$,+,$\cdot$), which I think is a ring; while in first-order theories, it is typical to see a theory defined like {0, 1, +, ·, =}, which I think is Peano arithmetic).

I see a close similarity between them, could anyone explain if they are the same or why not?

Answer 1

There are three distinct concepts here:

• A language $L$ (or "signature" or "similarity type" or "vocabulary" - these are all synonyms) is a collection of symbols naming distinguished elements and operations and relations. The language does not put any restriction on how these elements and operations and relation behave, it just specifies the symbols. Example: $L = \{0,1,+,\cdot\}$ is a language.
• A theory $T$ (in the language $L$) is a collection of axioms, i.e., sentences of first-order logic, using the symbols in $L$. Example: The ring axioms are a theory in the language $L$. Peano Arithmetic is another theory in the language $L$.
• A model (of a theory $T$) is a set equipped with interpretations of the symbols in the language $L$, which satisfies the sentences in $T$. Example: $\mathbb{Z}$, with the symbols $0$, $1$, $+$, and $\cdot$ interpreted in the usual ways, is a model of the ring axioms, but it is not a model of Peano Arithmetic. $\mathbb{N}$, with the symbols $0$, $1$, $+$, and $\cdot$ interpreted in the usual ways, is a model of Peano Arithmetic, but it is not a model of the ring axioms. We could also take $\mathbb{Z}$, interpret $+$ as multiplcation, interpret $\cdot$ as subtraction, interpret $0$ as $7$, and interpret $1$ as $42$. The resulting structure is neither a model of the ring axioms nor a model of Peano Arithmetic. But my point is that the symbols in the language $L$ can be interpreted completely arbitrarily, and it is up to the axioms in the theory to determine whether the resulting structure is a model.

I've just given you a very quick summary. For the precise definitions, pick up literally any introductory logic textbook that covers first-order logic.