Atomic model (mathematical logic)

<h2 id="definitions">Definitions</h2>
Let T be a <a href="/facts/Theory_(mathematical_logic)/mYe5gQpI">theory</a>. A complete type p(x1, ..., xn) is called principal or atomic (relative to T) if it is axiomatized relative to T by a single formula φ(x1, ..., xn) ∈ p(x1, ..., xn).
A formula φ is called complete in T if for every formula ψ(x1, ..., xn), the theory T ∪ {φ} entails exactly one of ψ and ¬ψ.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
It follows that a complete type is principal if and only if it contains a complete formula.
A model M is called atomic if every n-tuple of elements of M satisfies a formula that is complete in Th(M)—the theory of M.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Ordered_field/an0prSgG">ordered field</a> of <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a> is the unique atomic model of the theory of <a href="/facts/Real_closed_field/Vj1Un2fw">real closed fields</a>.</li>
<li>Any finite model is atomic.</li>
<li>A dense <a href="/facts/Linear_ordering/xnTtmuYJ">linear ordering</a> without endpoints is atomic.</li>
<li>Any <a href="/facts/Prime_model/Y15Ae4zv">prime model</a> of a <a href="/facts/Countable/Wj4DmWPv">countable</a> theory is atomic by the <a href="/facts/Type_(model_theory)/nuUHGFcg">omitting types theorem</a>.</li>
<li>Any countable atomic model is prime, but there are plenty of atomic models that are not prime, such as an uncountable dense linear order without endpoints.</li>
<li>The theory of a countable number of independent unary relations is complete but has no completable formulas and no atomic models.</li></ul>
<h2 id="properties">Properties</h2>
The <a href="/facts/Back-and-forth_method/XTWEYCXx">back-and-forth method</a> can be used to show that any two countable atomic models of a theory that are <a href="/facts/Elementarily_equivalent/3C94AfFq">elementarily equivalent</a> are <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</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), 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>
<li><a href="/facts/Wilfrid_Hodges/dgBdylPu">Hodges, Wilfrid</a> (1997), A shorter model theory, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-58713-6</li></ul>

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

<ol>
<li id="fn:1">Some authors refer to complete formulas as "atomic formulas", but this is inconsistent with the purely syntactical notion of an atom or atomic formula as a formula that does not contain a proper subformula. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

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Atomic model (mathematical logic)