Model complete theory

<h2 id="model-companion-and-model-completion">Model companion and model completion</h2>
A companion of a theory T is a theory T* such that every model of T can be embedded in a model of T* and vice versa.
A model companion of a theory T is a companion of T that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if T is an 
 
 
 
 
 ℵ
 
 0
 
 
 
 
 {\displaystyle \aleph _{0}}
 
-<a href="/facts/Categorical_theory/gTmQNu12">categorical theory</a>, then it always has a model companion.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
A model completion for a theory T is a model companion T* such that for any model M of T, the theory of T* together with the <a href="/facts/Diagram_(mathematical_logic)/N92uqsh2">diagram</a> of M is <a href="/facts/Complete_theory/TfBN8Fnb">complete</a>. Roughly speaking, this means every model of T is embeddable in a model of T* in a unique way.
If T* is a model companion of T then the following conditions are equivalent:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>T* is a model completion of T</li>
<li>T has the <a href="/facts/Amalgamation_property/TKzrzYx3">amalgamation property</a>.</li></ul>
If T also has universal axiomatization, both of the above are also equivalent to:

<ul><li>T* has <a href="/facts/Elimination_of_quantifiers/ftDaN3Y7">elimination of quantifiers</a></li></ul>
<h2 id="examples">Examples</h2>
<ul><li>Any theory with <a href="/facts/Elimination_of_quantifiers/ftDaN3Y7">elimination of quantifiers</a> is model complete.</li>
<li>The theory of <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed fields</a> is the model completion of the theory of fields. It is model complete but not complete.</li>
<li>The model completion of the theory of <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relations</a> is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements.</li>
<li>The theory of <a href="/facts/Real_closed_field/Vj1Un2fw">real closed fields</a>, in the language of <a href="/facts/Ordered_ring/4dGJVBIK">ordered rings</a>, is a model completion of the theory of <a href="/facts/Ordered_field/an0prSgG">ordered fields</a> (or even ordered <a href="/facts/Integral_domain/cylt6zxW">domains</a>).</li>
<li>The theory of real closed fields, in the language of <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>, is the model companion for the theory of <a href="/facts/Formally_real_field/IXsKBxEs">formally real fields</a>, but is not a model completion.</li></ul>
<h2 id="non-examples">Non-examples</h2>
<ul><li>The theory of dense linear orders with a first and last element is complete but not model complete.</li>
<li>The theory of <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a> (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.</li></ul>
<h2 id="sufficient-condition-for-completeness-of-model-complete-theories">Sufficient condition for completeness of model-complete theories</h2>
If T is a model complete theory and there is a model of T that embeds into any model of T, then T is complete.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Chen_Chung_Chang/vyMY21nb">Chang, Chen Chung</a>; <a href="/facts/Howard_Jerome_Keisler/FyQhCNS3">Keisler, H. Jerome</a> (1990) [1973]. Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-444-88054-3.</li></ul>
<ul><li><a href="/facts/Chen_Chung_Chang/vyMY21nb">Chang, Chen Chung</a>; <a href="/facts/Howard_Jerome_Keisler/FyQhCNS3">Keisler, H. Jerome</a> (2012) [1990]. Model Theory. Dover Books on Mathematics (3rd ed.). <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>. p. 672. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-48821-9.</li></ul>
<ul><li>Hirschfeld, Joram; Wheeler, William H. (1975). "Model-completions and model-companions". Forcing, Arithmetic, Division Rings. Lecture Notes in Mathematics. Vol. 454. Springer. pp. 44–54. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0064085">10.1007/BFb0064085</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-07157-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0389581">0389581</a>.</li></ul>
<ul><li>Marker, David (2002). Model Theory: An Introduction. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a> 217. New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98760-6.</li></ul>
<ul><li>Saracino, D. (August 1973). "Model Companions for ℵ0-Categorical Theories". <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>. 39 (3): 591–598.</li></ul>
<ul><li>Simmons, H. (1976). "Large and Small Existentially Closed Structures". <a href="/facts/Journal_of_Symbolic_Logic/lfXVXcoy">Journal of Symbolic Logic</a>. 41 (2): 379–390.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Saracino 1973. - Saracino, D. (August 1973). "Model Companions for ℵ0-Categorical Theories". Proceedings of the American Mathematical Society. 39 (3): 591–598. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Simmons 1976. - Simmons, H. (1976). "Large and Small Existentially Closed Structures". Journal of Symbolic Logic. 41 (2): 379–390. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Chang & Keisler 2012. - Chang, Chen Chung; Keisler, H. Jerome (2012) [1990]. Model Theory. Dover Books on Mathematics (3rd ed.). Dover Publications. p. 672. ISBN 978-0-486-48821-9. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Marker 2002. - Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. New York: Springer-Verlag. ISBN 0-387-98760-6. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Model complete theory open-in-new

Model complete theory