# Elementary embedding

Given two transitive structures $\mathcal{M}$ and $\mathcal{N}$, an elementary embedding from $\mathcal{M}$ to $\mathcal{N}$ is a function $j:\mathcal{M}\to\mathcal{N}$ such that $j(\mathcal{M})$ is an elementary substructure of $\mathcal{N}$, i.e. satisfies the same first-order sentences as $\mathcal{N}$ does. Obviously, if $\mathcal{M}=\mathcal{N}$, then $j(x)=x$ is an elementary embedding from $\mathcal{M}$ to itself, but is then called a trivial embedding. An embedding is nontrivial if there exists $x\in\mathcal{M}$ such that $j(x)\neq x$.

The critical point ($crit(j)$ or $cr(j)$) is the smallest ordinal moved by $j$. By $j$'s elementarity, $j(\kappa)$ must also be an ordinal, and therefore it is comparable with $\kappa$. It is easy to see why $j(\kappa)\leq\kappa$ is impossible, thus $j(\kappa)>\kappa$.

A prime model is one that embeds elementarily into every model of its theory.

## Definition

Given two transitive classes $\mathcal{M}$ and $\mathcal{N}$, and a function $j:\mathcal{M}\rightarrow\mathcal{N}$, $j$ is an elementary embedding if and only if for every first-order formula $\varphi$ with parameters $x_1,...,x_n\in\mathcal{N}$, one has: $$\mathcal{M}\models\varphi(x_1,...,x_2)\iff\mathcal{N}\models\varphi(j(x_1),...,j(x_2))$$

$j$ is nontrivial if and only if it has a critical point, i.e. $\exists\kappa(j(\kappa)\neq\kappa)$. Indeed, one can show by transfinite induction that if $j$ does not move any ordinal then $j$ does not move any set at all, thus a critical point must exists whenever $j$ is nontrivial.

## Tarski-Vaught Test

If $\mathcal{M}$ and $\mathcal{N}$ are both $\tau$-structures for some language $\tau$, and $j:\mathcal{M}\rightarrow\mathcal{N}$, then $j$ is an elementary embedding iff:

1. $j$ is injective (for any $x$ in $N$, there is at most one $y$ in $M$ such that $j(y)=x$).
2. $j$ has the following properties:
1. For any constant symbol $c\in\tau$, $j(c^\mathcal{M})=c^\mathcal{N}$.
2. For any function symbol $f\in\tau$ and $a_0,a_1...\in M$, $j(f^\mathcal{M}(a_0,a_1...))=f^\mathcal{N}(j(a_0),j(a_1)...)$. For example, $j(a_0+^\mathcal{M}a_1)=j(a_0)+^\mathcal{N}j(a_1)$.
3. For any relation symbol $r\in\tau$ and $a_0,a_1...\in M$, $r^\mathcal{M}(a_0,a_1...)\Leftrightarrow r^\mathcal{N}(j(a_0),j(a_1)...)$
3. For any first-order formula $\psi$ and any $x_0,x_1...\in M$ such that there is $y\in N$ with $\psi^\mathcal{N}(y,j(x_0),j(x_1)...)$, there is $z\in M$ with $\psi^\mathcal{M}(z,x_0,x_1...)$.

The Tarski-Vaught test refers to the special case where $\mathcal{M}$ is a substructure of $\mathcal{N}$ and $j(x)=x$ for every $x$.

This test determines if $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$. More specifically, $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$ iff for any $\psi$ and $x_0,x_1...\in M$ such that there is $y\in N$ with $\psi^\mathcal{N}(y,x_0,x_1...)$, there is $z\in M$ with $\psi^\mathcal{M}(z,x_0,x_1...)$.

## Use in Large Cardinal Axioms

There are two ways of making the critical point as large as possible:

1. Making $\mathcal{M}$ as large as possible, much larger than $\mathcal{N}$ (meaning that a "large" class can be embedded into a smaller class)
2. Making $\mathcal{M}$ and $\mathcal{N}$ more similar (for example, $\mathcal{M} = \mathcal{N}$ yet $j$ is nontrivial)

Using the first method, one can simply take $\mathcal{M}=V$ (the universe of all sets), and the resulting critical point is always a measurable cardinal, a very strong type of large cardinal, e.g. the first measurable is larger than infinitely many weakly compact cardinals (and much more).

Using the second method, one can take, say, $\mathcal{M} = \mathcal{N} = L$, i.e. create an embedding $j:L\to L$, whose existence has very important consequences, such as the existence of $0^\#$ (and thus $V\neq L$) and implies that every ordinal that is an uncountable cardinal in V is strongly inaccessible in L. By taking $\mathcal{M}=\mathcal{N}=V_\lambda$, i.e. a rank of the cumulative hiearchy, one obtains the very powerful rank-into-rank axioms, which sit near the very top of the large cardinal hiearchy. However, this second method has its limits, as shown by Kunen, as he showed that $\mathcal{M}=\mathcal{N}=V$ leads to an inconsistency with the axiom of choice, a theorem now known as the Kunen inconsistency. He also showed that a natural strengthening of the rank-into-rank axioms, $\mathcal{M}=\mathcal{N}=V_{\lambda+2}$ for some $\lambda\in Ord$, was inconsistent with the $AC$.

Most large cardinal axioms inbetween measurables and rank-into-rank axioms are obtained by mixing those two methods: one usually sets $\mathcal{M}=V$ then requires $\mathcal{N}$ to satisfies strong closure properties to make it "larger", i.e. closer to $V$ (that is, to $\mathcal{M}$). For example, $j:V\to\mathcal{N}$ is nontrivial with critical point $\kappa$ and the cumulative hiearchy rank $V_{j(\kappa)}$ is a subset of $\mathcal{N}$ then $\kappa$ is superstrong; if $\mathcal{N}$ contains all sequences of elements of $\mathcal{N}$ of length $\lambda$ for some $\lambda>\kappa$ then $\kappa$ is $\lambda$-supercompact, and so on.

The existence of a nontrivial elementary embedding $j:\mathcal{M}\to\mathcal{N}$ that is definable in $\mathcal{M}$ implies that the critical point $\kappa$ of $j$ is measurable in $\mathcal{M}$ (not necessarily in $V$). Every measurable ordinal is weakly compact and (strongly) inaccessible therefore its existence in any model is beyond $ZFC$, meaning that $ZFC$ cannot prove that such an embedding exists.

Here are some types of cardinals whose definition uses elementary embeddings:

The wholeness axioms also asserts the existence of elementary embeddings, though the resulting larger cardinal has no particular name. Vopěnka's principle is about elementary embeddings of set models.

## Absoluteness

The following results can be used in theorems about remarkable cardinals and other virtual variants.

(section from [1] unless otherwise noted)

The existence of an embedding of a countable model into another fixed model is absolute:

• For a countable first-order structure $M$ and an elementary embedding $j : M → N$, if $W ⊆ V$ is a transitive (set or class) model of (some sufficiently large fragment of) ZFC such that $M$ is countable in $W$ and $N ∈ W$, then $W$ has some elementary embedding $j^∗ : M → N$.
• If additionally both $M$ and $N$ are transitive $∈$-structures, we can assume that $crit(j^∗) = crit(j)$.
• We can also require that $j$ and $j^∗$ agree on some fixed finite number of values.

Therefore an elementary embedding $j : B → A$ between first-order structures exists in some set-forcing extension iff it already exists in $V^{Coll(ω,B)}$. Specifically, the following are equivalent for structures $B$ and $A$:

• There is a complete Boolean algebra $\mathbb{B}$ such that
$V^\mathbb{B} \models$ “There is an elementary embedding $j : B → A$.”
• In $V^{Coll(ω,B)}$ there is an elementary embedding $j : B → A$.
• For every complete Boolean algebra $\mathbb{B}$,
$V^\mathbb{B} \models$ “$|B| = \aleph_0 \implies$ There is an elementary embedding $j : B → A$.”

Moreover, if $B$ and $A$ are transitive $∈$-structures, we can assume that the embeddings have the same critical point and agree on finitely many fixed values.

These are also equivalent to player II having a winning strategy in game $G(B, A)$ defined in the following subsection.

Next fact: For first-order structures $M$ and $N$ in a common language, if there is an elementary embedding $j : M → N$ in some set-forcing extension, then there is such an embedding $j^∗ : M → N$ in any forcing extension in which $M$ has been made countable. Moreover, one can arrange that $j^∗$ agrees with $j$ on any given finite set of values and that, if appropriate, $crit(j) = crit(j^*)$.[2]

### In the language of game theory

To every pair of structures B and A of the same type, one can associate a closed game $G(B, A)$ (variant of an Ehrenfeucht-Fraı̈ssé game) such that $B$ elementarily embeds into $A$ in $V^{Coll(ω,B)}$ precisely when a particular player has a winning strategy in that game. Namely:

The game $G(B, A)$ is a game of length $ω$, where player I starts by playing some $b_0 ∈ B$ and player II responds by playing $a_0 ∈ A$. Players I and II continue to alternate, choosing elements $b_n$ and $a_n$ from their respective structures at stage $n$ of the game. II wins if for every $(n+1)$-parameter formula $φ$

$B \models φ(b_0 , . . . , b_n ) \iff A \models φ(a_0 , . . . , a_n)$

and I wins otherwise.

Since if II loses he must do so at some finite stage of the game, the game $G(B, A)$ is closed and hence determined by the Gale-Stewart theorem (Gale and Stewart, 1953). Thus, either I or II has a winning strategy.

Player II has a winning strategy precisely when $B$ elementarily embeds into $A$ in $V^{Coll(ω,B)}$. It follows that each first-order fragment of Generic Vopěnka’s Principle is characterised by the existence of certain winning strategies in its associated class of closed games.

## References

1. Bagaria, Joan and Gitman, Victoria and Schindler, Ralf. Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom. Arch Math Logic 56(1-2):1--20, 2017. www   DOI   MR   bibtex
2. Gitman, Victoria and Hamkins, Joel David. A model of the generic Vopěnka principle in which the ordinals are not Mahlo. , 2018. arχiv   bibtex
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