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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 $M \to N$ 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 $M(\varphi)$).

• 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 $f\colon M\to N$ 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.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeNov 23rd 2020

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?

1. Left a comment about a possibly confusing formulation.

Peter Arndt

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