Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)