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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.
And created large cardinal as a stub, mostly so that elementary embedding wouldn't be an orphan.
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.
Couldn't hurt, I think.
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 $M(\varphi)$).
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:
…
I’m confused by this passage in elementary embedding:
$\array{[\varphi]_M & \overset{}{\to} & [\varphi]_N\\ \downarrow && \downarrow\\ M A_1 \times\dots \times M A_n & \underset{}{\to} & N A_1 \times \dots \times N A_n}$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:
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.
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?
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