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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?

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  • $\begingroup$ What is their definition of the submodel generated by $X$? The smallest submodel that contains $X$? $\endgroup$ – Eric Wofsey Oct 25 '19 at 19:31
  • $\begingroup$ 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.) $\endgroup$ – Noah Schweber Oct 25 '19 at 19:45
  • $\begingroup$ 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 $\endgroup$ – mizejonathan17 Oct 25 '19 at 19:46
  • $\begingroup$ @NoahSchweber my reaction was that it should be B (superset symbol) {... as opposed to B = {... $\endgroup$ – mizejonathan17 Oct 25 '19 at 19:52
  • $\begingroup$ @NoahSchweber I get this but why then does it say “B = ...”? $\endgroup$ – mizejonathan17 Oct 25 '19 at 19:56
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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 $X$, $B\subseteq S$.

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