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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeNov 16th 2009
   • (edited Nov 16th 2009)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownCreated [[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. <latex>V\to V</latex>, and its use of choice, is unnecessary if you merely care about *definable* e.e.'s.

   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. , and its use of choice, is unnecessary if you merely care about definable e.e.'s.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 16th 2009
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownAnd created [[large cardinal]] as a stub, mostly so that [[elementary embedding]] wouldn't be an orphan.

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

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 16th 2009
   • PermaLink
   Author: David_Corfield
   Format: HtmlAre these embeddings the same as those we spoke about in the Fraisse limits post, or is 'elementary' qualifying somehow?

   Are these embeddings the same as those we spoke about in the Fraisse limits post, or is 'elementary' qualifying somehow?
4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 16th 2009
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   Author: Mike Shulman
   Format: MarkdownThe embeddings we used for Fraisse limits were only required to preserve and reflect the truth of *atomic* statements, rather than all first-order statements.

   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.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 16th 2009
   • PermaLink
   Author: David_Corfield
   Format: HtmlThanks. Would it be worth a page differentiating types of embedding - geometric, Yoneda? Or isn't there enough in common to merit such a page?

   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?
6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 16th 2009
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   Author: Mike Shulman
   Format: MarkdownCouldn't hurt, I think.

   Couldn't hurt, I think.

7. • CommentRowNumber7.
   • CommentAuthorjesse
   • CommentTimeMay 27th 2017
   • (edited May 27th 2017)
   • PermaLink
   Author: jesse
   Format: MarkdownItexI'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 $M \to N$ between models to be compatible with the models' interpretations of formulas in the theory, then the restriction (resp. pullback along $N(\varphi) \hookrightarrow N$) of your map to the interpretation $M(\varphi)$ of a formula $\phi$ in $M$ should land in the subobject $N(\varphi) \hookrightarrow N$ (resp. should come from $M(\varphi)$).

   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 $M \to N$ between models to be compatible with the models’ interpretations of formulas in the theory, then the restriction (resp. pullback along $N(\varphi) \hookrightarrow N$) of your map to the interpretation $M(\varphi)$ of a formula $\phi$ in $M$ should land in the subobject $N(\varphi) \hookrightarrow N$ (resp. should come from $M(\varphi)$).

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. 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: >* [[language]], [[signature]] >* [[theory]], [[first-order theory]] >* [[model]] >* [[interpretation]] >* [[structure in model theory]] >* [[elementary embedding]] > ...

   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:

   • language, signature

   • theory, first-order theory

   • model

   • interpretation

   • structure in model theory

   • elementary embedding

   …

9. • CommentRowNumber9.
   • CommentAuthorJohn Baez
   • CommentTimeNov 22nd 2020
   • (edited Nov 22nd 2020)
   • PermaLink
   Author: John Baez
   Format: MarkdownItexI'm confused by this passage in [[elementary embedding]]: > More generally, when we consider structures in a [[category]] as in [[categorical logic]], a morphism $f\colon M\to N$ between structures in $C$ is an **elementary embedding** if for any formula $\varphi$, the following square is a [[pullback]]: $$ \array{[\varphi]_M & \overset{}{\to} & [\varphi]_N\\ \downarrow && \downarrow\\ M A_1 \times\dots \times M A_n & \underset{}{\to} & N A_1 \times \dots \times N A_n} $$ > where $M A_i$ denotes the object of $C$ interpreting the type $A_i$, and $[\varphi]_M$ denotes the corresponding subobject in $C$ interpreting the truth value of the formula $\varphi$. One problem is that the types $A_i$ appear "out of the blue" - we start by saying "$f\colon M\to N$ between structures in $C$ is an **elementary embedding** if for any formula $\varphi$...", and none of this mentions types $A_i$. Can we add something to explain what they are, and what the $[\varphi]_M$ are, and what the maps $[\varphi]_M \to M A_1 \times\dots \times M A_n $ are? I'd do this myself if I could. <a href="https://ncatlab.org/nlab/revision/diff/elementary+embedding/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/elementary+embedding/15">v15</a>, <a href="https://ncatlab.org/nlab/show/elementary+embedding">current</a>

   I’m confused by this passage in elementary embedding:

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

   $$\array{[\varphi]_M & \overset{}{\to} & [\varphi]_N\\ \downarrow && \downarrow\\ M A_1 \times\dots \times M A_n & \underset{}{\to} & N A_1 \times \dots \times N A_n}$$

   where $M A_i$ denotes the object of $C$ interpreting the type $A_i$, and $[\varphi]_M$ denotes the corresponding subobject in $C$ interpreting the truth value of the formula $\varphi$.

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

   diff, v15, current

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2020
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexIt seems that the $A_i$ are the types implicitly referred to earlier: > parameters $a_1,\dots,a_n\in M$ (of appropriate [[type]]s) So this is typed first-order logic?

    It seems that the $A_i$ are the types implicitly referred to earlier:

    parameters $a_1,\dots,a_n\in M$ (of appropriate types)

    So this is typed first-order logic?

11. • CommentRowNumber11.
    • CommentAuthornLab edit announcer
    • CommentTimeNov 24th 2020
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    Author: nLab edit announcer
    Format: MarkdownItexLeft a comment about a possibly confusing formulation. Peter Arndt <a href="https://ncatlab.org/nlab/revision/diff/elementary+embedding/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/elementary+embedding/16">v16</a>, <a href="https://ncatlab.org/nlab/show/elementary+embedding">current</a>

    Left a comment about a possibly confusing formulation.

    Peter Arndt

    diff, v16, current