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Let $T$ be a first-order theory and suppose that $T$ can by axiomatised by a collection of positive sentences and that $T$ can axiomatised by a collection of $\Pi_2$-sentences. May we then conclude that $T$ can be axiomatised by a collection of positive $\Pi_2$-sentences?

Note that the above is true if we replace $\Pi_2$ with $\Pi_1$.

Lyndon's Theorem tells us that a first-order theory $T$ can be axiomatised by positive sentence precisely when its class of models $Mod(T)$ is closed under surjective homomorphisms. Similarly, by the Chang-Łoś-Suszko Theorem a first-order theory $T$ can be axiomatised by $\Pi_2$-sentences precisely when $Mod(T)$ is closed under the formation of chain colimits.

Thus I wonder if the classes of structures which enjoys both of these properties are precisely the classes of structures of the form $Mod(T)$, for some first-order theory $T$ consisting of positive $\Pi_2$-sentences.

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    Do you have a reference or easy proof for the $\Pi_1$ case? – 2017-02-20
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    The $\Pi_1$ case is stated in Lyndon's "Properties preserved under algebraic constructions" (1959), p. 292. However, look at this reference again I see that no proof is given. – 2017-02-20
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    I've just checked Chang & Keisler. They give the $\Pi_1$ case as an exercise (Exercise 3.2.2). And there's no *, so it's probably not too hard (I haven't tried to prove it). There's no mention of the $\Pi_2$ case. – 2017-02-20

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