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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 16th 2009
    • (edited Nov 16th 2009)

    Created elementary embedding to record a proof that I learned from Nate Ackermann. The upshot is that Kunen's technical argument for the inconsistency of an e.e. V\to V, and its use of choice, is unnecessary if you merely care about definable e.e.'s.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeNov 16th 2009

    And created large cardinal as a stub, mostly so that elementary embedding wouldn't be an orphan.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 16th 2009
    Are these embeddings the same as those we spoke about in the Fraisse limits post, or is 'elementary' qualifying somehow?
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 16th 2009

    The embeddings we used for Fraisse limits were only required to preserve and reflect the truth of atomic statements, rather than all first-order statements.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 16th 2009
    Thanks. Would it be worth a page differentiating types of embedding - geometric, Yoneda? Or isn't there enough in common to merit such a page?
    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeNov 16th 2009

    Couldn't hurt, I think.

    • CommentRowNumber7.
    • CommentAuthorjesse
    • CommentTimeMay 27th 2017
    • (edited May 27th 2017)

    I’ve formalized Urs’ observation from seven(!) years ago that an elementary embedding is a natural transformation, under Proposition 4.1. The following Remark 4.2 is closely related to Mike’s pullback condition: if you want a map MNM \to N between models to be compatible with the models’ interpretations of formulas in the theory, then the restriction (resp. pullback along N(φ)NN(\varphi) \hookrightarrow N) of your map to the interpretation M(φ)M(\varphi) of a formula ϕ\phi in MM should land in the subobject N(φ)NN(\varphi) \hookrightarrow N (resp. should come from M(φ)M(\varphi)).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017

    Thanks. I noticed that an entry “elementary embedding” was missing from model theory - contents and so I added it. In the course of this I rearranged slightly the list of keywords at the beginning, trying to put them more in logical order. Now it starts out like this:

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeNov 22nd 2020
    • (edited Nov 22nd 2020)

    I’m confused by this passage in elementary embedding:

    More generally, when we consider structures in a category as in categorical logic, a morphism f:MNf\colon M\to N between structures in CC is an elementary embedding if for any formula φ\varphi, the following square is a pullback:

    [φ] M [φ] N MA 1××MA n NA 1××NA n \array{[\varphi]_M & \overset{}{\to} & [\varphi]_N\\ \downarrow && \downarrow\\ M A_1 \times\dots \times M A_n & \underset{}{\to} & N A_1 \times \dots \times N A_n}

    where MA iM A_i denotes the object of CC interpreting the type A iA_i, and [φ] M[\varphi]_M denotes the corresponding subobject in CC interpreting the truth value of the formula φ\varphi.

    One problem is that the types A iA_i appear “out of the blue” - we start by saying “f:MNf\colon M\to N between structures in CC is an elementary embedding if for any formula φ\varphi…”, and none of this mentions types A iA_i. Can we add something to explain what they are, and what the [φ] M[\varphi]_M are, and what the maps [φ] MMA 1××MA n[\varphi]_M \to M A_1 \times\dots \times M A_n are? I’d do this myself if I could.

    diff, v15, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2020

    It seems that the A iA_i are the types implicitly referred to earlier:

    parameters a 1,,a nMa_1,\dots,a_n\in M (of appropriate types)

    So this is typed first-order logic?

  1. Left a comment about a possibly confusing formulation.

    Peter Arndt

    diff, v16, current