Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 23 / 3 / c

Table of contents
- Solution c

Solution

A model companion

T^{*}

of a first-order theory

T

, in the same first-order language, is a model-complete theory with the same universal consequences of a theory as

T

:

T^{*} is model-complete and T_{\forall}^{*} = T_{\forall} .

Equivalently, every model of

T

embeds into a model of

T^{*}

, and every model of

T^{*}

embeds into a model of

T

. The equivalence follows from the diagram embedding criterion for universal theories, which is an application of the compactness theorem. A model companion need not be a syntactic extension of

T

, and uniqueness is understood up to logical equivalence of theories.

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