Thanks for the reminder! It’s now on my to-do list.

]]>Jesse,

with all the nice material that you are adding on model theory. Might you have the energy to expand the (presently puny) table of contents at

?

(This entry is the one that is included as the “floating table of contents” in the top right of all the model theory entries.)

As an example for what I have in mind, compare to

]]>I added a remark (currently Remark 4.1) to diagram of a first-order structure that passing from $T$ to $T_{\mathsf{Diag(M)}}$ (resp. $\mathsf{EDiag}$) is functorial on interpretations the way that one would expect.

]]>For the sake of having a reference to link to later, I’ve written diagram of a first-order structure. This is just a construction where you take a theory $T$ and expand it to a new theory $T'$ by naming one of its models $M$ with constant symbols for each element of $M$ while additionally stipulating those constant symbols have to behave like they came from $M$.

If those additional stipulations were only quantifier-free, the models of $T'$ are those models of $T$ containing $M$ as an induced substructure.

If those additional stipulations were all the first-order sentences satisfied by the elements of $M$, then the models of $T'$ are those models of $T$ containing $M$ as an elementary substructure.

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