Is Hodges’ definition of elementary embedding wrong

first-order-logiclogicmodel-theoryreference-request

I am reading the beautiful book by Hodges, A Shorter Model Theory. In sections 2.4 and 2.5 the authors classifies formulas through maps which preserve them and viceversa. All is very clear, except from the definition of elementary embedding.
In fact, given two $L$-structures $A,B$ Hodges says:

  1. An homomorphism $f:A\rightarrow B$ preserves a set of formulas $\Phi$ if $A\models\phi[\bar{a}]\Rightarrow B\models\phi[f\bar{a}]\quad \forall\phi\in\Phi$
  2. An elementary embedding is an embedding $i:A\rightarrow B$ which preserves any first order formula
    $A\models\phi[\bar{a}]\Rightarrow B\models\phi[i\bar{a}]\quad \forall\phi$

The fact is, he seems to use another definition of elementary embedding, which is the one in 2, but with $\Leftrightarrow$. This is also the one I know. In particular if we then consider elementary substructures, I cannot see how this definition can work, since basically in elementary substructures we have formulas preserved in both directions yielding elementary equivalence. Moreover under this definition any isomorphism should be an elementary embedding, which is not the case, considering $A=(\mathbb{Z},<), B=(2\mathbb{Z},<)$

I am feeling very stupid not seeing why these two approaches are equivalent, but I also checked the errata and nothing is said about this, so I guess it should be.

Best Answer

This $\Longrightarrow$ implies the desired $\Longleftrightarrow$. It would indeed make more sense to define an elementary embedding $f$ to be a map $f : A \mapsto B$ satisfying $\forall \phi \ \forall \bar{a} \in A, \, A\models\phi[\bar{a}]\Longleftrightarrow B\models\phi[f\bar{a}]$ (Btw notice that this implies that $f$ is injective). Then, you can make the following observation :

Claim Assume a map $f : A \mapsto B$ satisfies $\forall \phi \ \forall \bar{a} \in A, \, A\models\phi[\bar{a}]\Longrightarrow B\models\phi[f\bar{a}]$. Then $f$ is an elementary embedding.

Let $f$ be as in claim, we show that $f$ is an elementary embedding :
All what remains to do is to show that if $B \models \phi[f\bar{a}]$ for some $\phi, \bar{a}$, then $A\models \phi[\bar{a}]$. Assume for contradiction that this does not hold, then $A \models \neg \phi[\bar{a}]$ and by assumption we get $B \models \neg \phi[f\bar{a}]$, a contradiction to our assumption that $B\models \phi[f\bar{a}]$ !

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