What is the point of expanding a structure $M$ by constants from $M$ itself?

Question

In model theory, the expansion of $L$ by constants from $D$, in symbols $L(D)$, is, by definition, the uniquely determined language of the signature $(C\cup D,F,R,\sigma')$. In particular, often times, we have an underlying set, say $M$, and we take constants from $M$, say $A$, to expand our language as we did above, and then talk about the expanded structure $(M,A)$.

My question is, how does this expand anything if we are taking constants from $M$, and what is the use of this construction?

Any help would be appreciated.

Cheers!

Answer 1

“...how does this expand anything if we are taking constants from $M$...” you are expanding your language, for example if your language is the language of group theory and you have a structure that satisfies the group axioms, you add a new constant symbol for every element in the group. So previously you only had $e$ as a constant symbol in the language, now you have a lot new constant symbols.

Using this construction, for example, you can talk about “complete diagrams”, the set of sentences that the original structure with the new constants interpreted in the usual way satisfies (you can think about this as the “multiplication table” in some sense). Without adding new constant symbols, you cannot talk about this set, so adding the new constants gives you a new degree of freedom in some sense.

Answer 2

Adding constants for elements of a structure serves mostly to allow us to use them (elements) as parameters in sentences in a uniform way.

Thus, for any $M$, tuple $\bar a$ in $M$ and formula $\varphi(\bar x,\bar y)$, $\varphi(\bar x,\bar a)$ is still just a plain formula (in $L(\bar a)\subseteq L(M)$). Formulas with parameters is frequently useful, and this allows us to treat them just the same as all the other formulas (just in a slightly different language).

When we expand the language by constants in a set $A\subseteq M$, we typically also expand the theory to include some properties of the constants, at the very least adding their quantifier-free properties, which is called the atomic diagram of $A$, or all their elementary properties, which is called the elementary diagram of $A$. Both of these are theories in $L(A)$.

In the specific case when $A=M$ is a model, the models of the atomic diagram of $M$ are exactly the structures containing $M$ as a substructure, and the models of the elementary diagram are exactly those which contain $M$ as an elementary substructure.