A first-order logic is interpreted in a model where sentences of the logic can be said to be true or false. There may be more than one model, and we can identify morphisms between models.

Do we have a universal model, that is one in which all other models are quotients of this one?

If not, are there nice conditions under which there is?

For example, for geometric theories, we have the classifying topos:

So the classifying topos for the geometric theory $T$ is a Grothendieck topos $S[T]$ equipped with a “universal model $U$ of $T$”. This means that for any Grothendieck topos $E$ together with a model $X$ of $T$ in $E$, there exists a unique (up to isomorphism) geometric morphism $f:E \rightarrow S[T]$ such that $f^*$ maps the $T$-model $U$ to the model $X$.


By the Upward Lowenheim-Skolem Theorem, any first-order theory that has an infinite model has models of arbitrarily large cardinality. So there cannot be a universal model.

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    $\begingroup$ We can also just explicitly exhibit a counterexample, e.g. the first-order theory of groups. $\endgroup$ – Qiaochu Yuan Aug 11 '13 at 17:21
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    $\begingroup$ The nature of the universal model in the sense of classifying toposes is very different, though... $\endgroup$ – Zhen Lin Aug 13 '13 at 11:13

As far as a "nice" class of first-order theories go, there is the subject matter of varieties in universal algebra, where one has essentially axioms for operations and constants with identities (equational), e.g. groups, rings, and boolean algebras.

Here one can "generate" all the models of the theory by taking quotients, subalgebras, and products of varieties.

The Question here asks if in some cases the models might all be generated by taking quotients of a single model (no). See however Birkhoff's HSP Theorem, according to which any class closed under quotients (homomorphisms), subalgebras, and products must be the models of some equational theory.


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