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    • CommentRowNumber1.
    • CommentAuthorjesse
    • CommentTimeMay 8th 2017

    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 TT and expand it to a new theory TT' by naming one of its models MM with constant symbols for each element of MM while additionally stipulating those constant symbols have to behave like they came from MM.

    If those additional stipulations were only quantifier-free, the models of TT' are those models of TT containing MM as an induced substructure.

    If those additional stipulations were all the first-order sentences satisfied by the elements of MM, then the models of TT' are those models of TT containing MM as an elementary substructure.

    • CommentRowNumber2.
    • CommentAuthorjesse
    • CommentTimeMay 19th 2017
    • (edited May 19th 2017)

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

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2017
    • (edited May 19th 2017)

    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

    • CommentRowNumber4.
    • CommentAuthorjesse
    • CommentTimeMay 19th 2017

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