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model complete theory

Contents

Idea

Model completeness is a property of first-order theories (in the same way that existential closedness is a property of first-order structures) meant to generalize the properties of the theory ACF of algebraically closed fields.

As discussed in Hodges’ Model theory:

In the early 1950s Abraham Robinson noticed that certain maps studied by algebraists are in fact elementary embeddings. If you choose a map at random, the chances of it being an elementary embedding are negligible. So Robinson reckoned that there must be a systematic reason for the appearance of these elementary embeddings, and he set out to find what the reason was.

In the course of this quest he introduced the notions of model complete theories, companionable theories and model companions?. These notions have become essential tools for the model theory of algebra.

Definition

We say a first-order theory TT is model complete if every embedding (not necessarily elementary) between models of TT is elementary.

The following characterization appears as Theorem 8.3.1 in (Hodges93):

Theorem

Let TT be as above. The following are equivalent:

  1. TT is model complete.

  2. Every model of TT is existentially closed.

  3. For every embedding ABA \hookrightarrow B where AA and BB model TT, there exists an elementary extension DD of AA and an embedding g:BDg : B \to D such that the triangle commutes.

  4. Every existential formula is equivalent (mod TT) to a universal formula.

  5. Every formula is equivalent (mod TT) to a universal formula.

Examples

Remarks

  • A theory TT is model complete if and only if for every model AA of TT, the quantifier-free diagram T Diag(A)T_{\mathsf{Diag}(A)} (whose models are precisely the models of TT containing AA as an embedded substructure) is complete.

  • Analogously, a theory TT is substructure complete if and only if for every model AA of TT and every substructure BAB \subseteq A, the quantifier-free diagram T Diag(B)T_{\mathsf{Diag}(B)} is complete.

  • A model complete theory is an “existentially closed theory” in the sense that all of its models are existentially closed models.

  • As mentioned above, substructure completeness implies model completeness because every elementary submodel is also a substructure. The proposition below shows that the converse holds with some additional assumptions.

Model completeness + amalgamation property implies substructure completeness

Proposition

Suppose the first-order theory TT in the language \mathcal{L} is model complete and has the property that for any two models X,YTX,Y \models T and a mutual \mathcal{L}-substructure AX,YA \hookrightarrow X,Y, the latter span in the category of \mathcal{L}-structures and embeddings has a cocone ZZ which is also a model of TT.

Then TT is substructure complete.

This property of being able to “amalgamate” the models XX and YY over the common substructure AA is sometimes called the amalgamation property, though this terminology is rather overloaded (c.f. Fraisse limit).

Proof

Let AA be a substructure of some model MTM \models T. Append the quantifier-free diagram of AA to TT to form T Diag(A)T_{\mathsf{Diag}(A)}. Suppose towards a contradiction that φ\varphi is a sentence which is undecided by T Diag(A)T_{\mathsf{Diag}(A)}; let XX and YY be models of T Diag(A)T_{\mathsf{Diag}(A)} which witness this, so that XφX \models \varphi while Y¬φY \models \neg \varphi.

Let ZZ amalgamate XX and YY over AA.

By model completeness, the theory T Diag(X)T_{\mathsf{Diag}(X)} of TT-models (not necessarily elementarily) embedding XX is complete. Abuse notation and replace the instances of constants from AA in φ\varphi with their images in XX. Either T Diag(X)φT_{\mathsf{Diag}(X)} \models \varphi or T Diag(X)¬φT_{\mathsf{Diag}(X)} \models \neg \varphi. Since XX certainly embeds itself and XφX \models \varphi, T Diag(X)φT_{\mathsf{Diag}(X)} \models \varphi.

Similarly, T Diag(Y)¬φ.T_{\mathsf{Diag}(Y)} \models \neg \varphi.

Now, on one hand, the theory T Diag(X)T Diag(Y)/AT_{\mathsf{Diag}(X)} \cup T_{\mathsf{Diag}(Y)}/A (where we glue along the copies of AA in Diag(X)\mathsf{Diag}(X) and Diag(Y)\mathsf{Diag}(Y)) entails both φ\varphi and ¬φ\neg \varphi, which makes it inconsistent. This theory therefore has no models.

On the other hand, this theory axiomatizes the class of amalgams of XX and YY over AA in the category of \mathcal{L}-structures and embeddings, so in particular is modelled by ZZ, a contradiction.

References

  • Wilfrid Hodges, Model theory, Cambridge Univ. Press 1993

Last revised on May 27, 2017 at 02:34:03. See the history of this page for a list of all contributions to it.