# C.C. Chang’s Explanation of Generated Submodels

I have no issue with Chang and Keisler’s definition of “submodels generated by...” but I’m extremely confused with how they go on to define the universe of the certain submodel $$B$$ generated by $$X$$ (a nonempty subset of $$A$$, a model for the language $$L$$). They state the following:

$$B = \{t[x_1,\dots,x_n] : t\text{ is a term of L and }x_1,\dots,x_n\in X\}$$

Why must this be so?

• What is their definition of the submodel generated by $X$? The smallest submodel that contains $X$? – Eric Wofsey Oct 25 '19 at 19:31
• Is there an example you have in mind where you think it should be otherwise? (Note that $t$ is allowed to be any term, not just one of the functions in the language, and we can "compose" terms.) – Noah Schweber Oct 25 '19 at 19:45
• So they define a “submodel generated by X” in the following way: Let A be a model for L and let X be a nonempty subset of A. Let B = intersection symbol {C : C (model) subset of symbol A and X subset of C} Then there is a submodel B subset of A with universe B. B is called the submodel generated by X – mizejonathan17 Oct 25 '19 at 19:46
• @NoahSchweber my reaction was that it should be B (superset symbol) {... as opposed to B = {... – mizejonathan17 Oct 25 '19 at 19:52
• @NoahSchweber I get this but why then does it say “B = ...”? – mizejonathan17 Oct 25 '19 at 19:56

Let $$S=\{t[x_1,\dots,x_n] : t\text{ is a term of L and }x_1,\dots,x_n\in X\}$$. Clearly any submodel $$C$$ of $$A$$ that contains $$X$$ must also contain $$S$$, since every element of $$S$$ is obtained by repeatedly applying operations starting from elements of $$X$$ and $$C$$ must be closed under the operations. (More formally, you can prove by induction on terms that each element of $$S$$ is in $$C$$.) So, $$S\subseteq B$$.
To prove the reverse inclusion, simply observe that $$S$$ itself is a submodel of $$A$$: it is closed under all the operations since if you apply an operation to some terms you just get a bigger term using the function symbol corresponding to the operation. So $$S$$ is a submodel of $$A$$ that contains $$X$$, $$B\subseteq S$$.