# Reference request: quantifier elimination test

I'm having difficulty finding this result in the standard texts.

Theorem. Let $$T$$ be a theory in a language $$\mathcal{L}$$. TFAE:

1) $$T$$ has quantifier elimination,

2) Whenever $$M, N$$ are $$\aleph_1$$-saturated models of $$T$$, $$A \subset M$$, $$B \subset N$$ are countable nonempty substructures and $$f : A \rightarrow B$$ an $$\mathcal{L}$$-isomorphism, then for any $$a \in M$$ there exists an extension $$f' \supset f$$ with $$a$$ in the domain of $$f'$$ and $$f'$$ still an $$\mathcal{L}$$-isomorphism. In addition, for any $$b \in N$$ there exists an extension $$f'' \supset f$$ with $$b$$ in the range of $$f''$$ and $$f''$$ still an $$\mathcal{L}$$-isomorphism.

I'm obtaining this result from these notes by Chatzidakis (see 2.27): http://www.math.ens.fr/~zchatzid/papiers/CUP-MT.pdf. Any suggestions are appreciated.

• You really mean to replace $\aleph_1$-saturated by $|{\mathcal L}|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped. Jul 7 '19 at 18:15
• You are correct, I had a particular example in mind while typing and did not adjust for that. Thank you! Aug 25 '19 at 17:28

I'm not sure if this formulation with $$\omega_1$$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).
Let $$T$$ be an $$L$$-theory. Suppose that for all quantifier-free formulas $$\phi(\bar{v},w)$$, if $$M,N\models T$$, $$A$$ is a common substructure of $$M$$ and $$N$$, $$\bar{a}\in A$$, and there is $$b\in M$$ such that $$M\models\phi(\bar{a},b)$$, then there is $$c\in N$$ such that $$N\models \phi(\bar{a},c)$$. Then $$T$$ has quantifier elimination.