SherbyNet - /page/wikibook/entry.mu

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<title>Abstract elementary class</title>

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Abstract elementary class

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Model_theory" title="Model theory">model theory</a>, a discipline within <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an <a href="Elementary_substructure" class="mw-redirect" title="Elementary substructure">elementary substructure</a> of an <a href="Elementary_class" title="Elementary class">elementary class</a> in <a href="First-order_logic" title="First-order logic">first-order</a> model theory. They were introduced by <a href="Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a>.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle \langle K,\prec _{K}\rangle }</annotation>

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</math><img src="./a2a2f43699583bac94dc9bcf2b3f36c98fde29e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle \langle K,\prec _{K}\rangle }" loading="lazy">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> a class of structures in some language <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=L(K)}">

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<annotation encoding="application/x-tex">{\displaystyle L=L(K)}</annotation>

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</math><img src="./e478c9982edf1cec60b50f2eae95f6f6f5d05e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.139ex; height:2.843ex;" alt="{\displaystyle L=L(K)}" loading="lazy">, is an AEC if it has the following properties:

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<annotation encoding="application/x-tex">{\displaystyle \prec _{K}}</annotation>

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</math><img src="./02cb525bfd1752d14a61117fadd82ab235b8bcf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.501ex; height:2.176ex;" alt="{\displaystyle \prec _{K}}" loading="lazy"> is a <a href="Partial_order" class="mw-redirect" title="Partial order">partial order</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle M\prec _{K}N}</annotation>

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</math><img src="./04a58ebb4bda1286c0751a580301f6705756bcf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.297ex; height:2.509ex;" alt="{\displaystyle M\prec _{K}N}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a substructure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">.</li>

<li>Isomorphisms: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is closed under <a href="Isomorphism" title="Isomorphism">isomorphisms</a>, and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N,M',N'\in K,}">

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<annotation encoding="application/x-tex">{\displaystyle M,N,M',N'\in K,}</annotation>

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</math><img src="./d8788c65fcafc391c0d229ef2bb2a130b1f203fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.152ex; height:2.843ex;" alt="{\displaystyle M,N,M',N'\in K,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\simeq M',}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon M\simeq M',}</annotation>

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</math><img src="./616244431be16c3f7e848b5e0b2a4eda89620da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.684ex; height:2.843ex;" alt="{\displaystyle f\colon M\simeq M',}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon N\simeq N',}">

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<annotation encoding="application/x-tex">{\displaystyle g\colon N\simeq N',}</annotation>

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</math><img src="./72622a2fa37a47ab49b897dfb4cd1bf4be589896.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.767ex; height:2.843ex;" alt="{\displaystyle g\colon N\simeq N',}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\subseteq g,}">

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<annotation encoding="application/x-tex">{\displaystyle f\subseteq g,}</annotation>

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</math><img src="./f211074f32302d828138fe5c78e09c3751553363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.14ex; height:2.509ex;" alt="{\displaystyle f\subseteq g,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\prec _{K}N,}">

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<annotation encoding="application/x-tex">{\displaystyle M\prec _{K}N,}</annotation>

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</math><img src="./0b185ac65fa4ee74a029763ce525135dcc2fb6b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.944ex; height:2.509ex;" alt="{\displaystyle M\prec _{K}N,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M'\prec _{K}N'.}">

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<annotation encoding="application/x-tex">{\displaystyle M'\prec _{K}N'.}</annotation>

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</math><img src="./794bcd70b2911c7bc6a9af28f4596a248d52bd69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.43ex; height:2.843ex;" alt="{\displaystyle M'\prec _{K}N'.}" loading="lazy"></li>

<li>Coherence: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}\prec _{K}M_{3},}">

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<annotation encoding="application/x-tex">{\displaystyle M_{1}\prec _{K}M_{3},}</annotation>

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</math><img src="./4b612de7c686e3df3f51355d225e79a9973f7cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:2.509ex;" alt="{\displaystyle M_{1}\prec _{K}M_{3},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}\prec _{K}M_{3},}">

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<annotation encoding="application/x-tex">{\displaystyle M_{2}\prec _{K}M_{3},}</annotation>

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</math><img src="./44ad8146ff4aada2a6bdb14739e83af3160b76e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:2.509ex;" alt="{\displaystyle M_{2}\prec _{K}M_{3},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}\subseteq M_{2},}">

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<annotation encoding="application/x-tex">{\displaystyle M_{1}\subseteq M_{2},}</annotation>

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</math><img src="./f892309672725f7c06c85b9050d4fac655e48b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.362ex; height:2.509ex;" alt="{\displaystyle M_{1}\subseteq M_{2},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}\prec _{K}M_{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle M_{1}\prec _{K}M_{2}.}</annotation>

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</math><img src="./a674ffda10922ea21cac2c772cc88a30266959d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:2.509ex;" alt="{\displaystyle M_{1}\prec _{K}M_{2}.}" loading="lazy"></li>

<li>Tarski–Vaught <a href="Total_order" title="Total order">chain</a> axioms: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is an <a href="Ordinal_number" title="Ordinal number">ordinal</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\,M_{\alpha }\mid \alpha <\gamma \,\}\subseteq K}">

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<annotation encoding="application/x-tex">{\displaystyle \{\,M_{\alpha }\mid \alpha <\gamma \,\}\subseteq K}</annotation>

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</math><img src="./f054f88286ea97ebd4b61ed67b24f206ff574981.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.587ex; height:2.843ex;" alt="{\displaystyle \{\,M_{\alpha }\mid \alpha <\gamma \,\}\subseteq K}" loading="lazy"> is a chain (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\beta <\gamma \implies M_{\alpha }\prec _{K}M_{\beta }}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha <\beta <\gamma \implies M_{\alpha }\prec _{K}M_{\beta }}</annotation>

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</math><img src="./50e35a2ea72dec35eafb95c27f234f6be5501520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.423ex; height:2.843ex;" alt="{\displaystyle \alpha <\beta <\gamma \implies M_{\alpha }\prec _{K}M_{\beta }}" loading="lazy">), then:

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<annotation encoding="application/x-tex">{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\in K}</annotation>

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</math><img src="./7240b859d41ef5cfa979777cf5763dccf9c8d3e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.055ex; height:5.843ex;" alt="{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\in K}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle M_{\alpha }\prec _{K}N}</annotation>

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</math><img src="./688e47510bc75fc852264e464c4fea1b1d023e6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.393ex; height:2.509ex;" alt="{\displaystyle M_{\alpha }\prec _{K}N}" loading="lazy">, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha <\gamma }</annotation>

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</math><img src="./0ef5c7601ba2d67261f1f13a6bff4d00581284bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:2.343ex;" alt="{\displaystyle \alpha <\gamma }" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\prec _{K}N}">

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<annotation encoding="application/x-tex">{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\prec _{K}N}</annotation>

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</math><img src="./fbeb9e48cdedc57d34b55299b6e24f8ab8fec6ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.004ex; height:5.843ex;" alt="{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\prec _{K}N}" loading="lazy"></li></ul></li>

<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a> axiom: There exists a <a href="Cardinal_number" title="Cardinal number">cardinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \geq |L(K)|+\aleph _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu \geq |L(K)|+\aleph _{0}}</annotation>

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</math><img src="./72fa99bbe8e7f72508e0b1584c0652e32a6eb792.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.567ex; height:2.843ex;" alt="{\displaystyle \mu \geq |L(K)|+\aleph _{0}}" loading="lazy">, such that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a subset of the universe of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, then there is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> whose universe contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|N\|\leq |A|+\mu }">

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<annotation encoding="application/x-tex">{\displaystyle \|N\|\leq |A|+\mu }</annotation>

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</math><img src="./f83719292d25de4f7ec22177b58ba1d6bc86bf86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.766ex; height:2.843ex;" alt="{\displaystyle \|N\|\leq |A|+\mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\prec _{K}M}">

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<annotation encoding="application/x-tex">{\displaystyle N\prec _{K}M}</annotation>

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</math><img src="./f804aa53b99ec108ba325c650d639072d569d5f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.297ex; height:2.509ex;" alt="{\displaystyle N\prec _{K}M}" loading="lazy">. We let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {LS} (K)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {LS} (K)}</annotation>

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</math><img src="./0b762e549c2ad12841c79e364ee9e55c9a289dae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.621ex; height:2.843ex;" alt="{\displaystyle \operatorname {LS} (K)}" loading="lazy"> denote the least such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and call it the Löwenheim–Skolem number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">.</li></ul>

Note that we usually do not care about the models of size less than the Löwenheim–Skolem number and often assume that there are none (we will adopt this convention in this article). This is justified since we can always remove all such models from an AEC without influencing its structure above the Löwenheim–Skolem number.

A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-embedding is a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\rightarrow N}">

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<annotation encoding="application/x-tex">{\displaystyle f:M\rightarrow N}</annotation>

</semantics>

</math><img src="./c6cd11d4656ac58a9d4c9ef09866d91950f6cc7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.336ex; height:2.509ex;" alt="{\displaystyle f:M\rightarrow N}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,N\in K}</annotation>

</semantics>

</math><img src="./00d7a51a71a769b425241f8e214e8c249b10075e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.446ex; height:2.509ex;" alt="{\displaystyle M,N\in K}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[M]\prec _{K}N}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f[M]\prec _{K}N}</annotation>

</semantics>

</math><img src="./d5023d23810a884e0fbf0547b94f76506b521f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.87ex; height:2.843ex;" alt="{\displaystyle f[M]\prec _{K}N}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is an isomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[M]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f[M]}</annotation>

</semantics>

</math><img src="./da22846ca5b315395b2541cc80646b450e7cb7f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.014ex; height:2.843ex;" alt="{\displaystyle f[M]}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is clear from context, we omit it.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

The following are examples of abstract elementary classes:<a href="#cite_note-2">[2]</a>

<ul><li>An <a href="Elementary_class" title="Elementary class">Elementary class</a> is the most basic example of an AEC: If T is a first-order theory, then the class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mod} (T)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Mod} (T)}</annotation>

</semantics>

</math><img src="./8d5c5d37aede9b468541cead570e82558db7fd3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.032ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mod} (T)}" loading="lazy"> of models of T together with <a href="Elementary_substructure" class="mw-redirect" title="Elementary substructure">elementary substructure</a> forms an AEC with Löwenheim–Skolem number |T|.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> is a sentence in the <a href="Infinitary_logic" title="Infinitary logic">infinitary logic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\omega _{1},\omega }}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }}</annotation>

</semantics>

</math><img src="./037b84dcf7eb85313f964632c939bce6464a657c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.149ex; height:2.843ex;" alt="{\displaystyle L_{\omega _{1},\omega }}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> is a countable <a href="Fragment_(logic)" title="Fragment (logic)">fragment</a> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \operatorname {Mod} (T),\prec _{\mathcal {F}}\rangle }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \operatorname {Mod} (T),\prec _{\mathcal {F}}\rangle }</annotation>

</semantics>

</math><img src="./f01340f12cce5800d9e50991d31a3c544aca6eda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.277ex; height:2.843ex;" alt="{\displaystyle \langle \operatorname {Mod} (T),\prec _{\mathcal {F}}\rangle }" loading="lazy"> is an AEC with Löwenheim–Skolem number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation>

</semantics>

</math><img src="./721cd7f8c15a2e72ad162bdfa5baea8eef98aab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}" loading="lazy">. This can be generalized to other logics, like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\kappa ,\omega }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{\kappa ,\omega }}</annotation>

</semantics>

</math><img src="./5fdf99b906580bf5256ad5902cffe10079e61d33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.242ex; height:2.843ex;" alt="{\displaystyle L_{\kappa ,\omega }}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\omega _{1},\omega }(Q)}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }(Q)}</annotation>

</semantics>

</math><img src="./3d448f1a6cdca64d74c8beeba71dd8dd5f6f92eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.796ex; height:3.009ex;" alt="{\displaystyle L_{\omega _{1},\omega }(Q)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

</semantics>

</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> expresses "there exists uncountably many".</li>

<li>If T is a first-order countable <a href="Superstable" class="mw-redirect" title="Superstable">superstable</a> theory, the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation>

</semantics>

</math><img src="./78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}" loading="lazy">-saturated models of T, together with elementary substructure, is an AEC with Löwenheim–Skolem number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}}">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}}</annotation>

</semantics>

</math><img src="./779da5db4ed54fa334dd92089cdf1c284e45febb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.231ex; height:2.676ex;" alt="{\displaystyle 2^{\aleph _{0}}}" loading="lazy">.</li>

<li><a href="Schanuel's_conjecture#Zilber's_pseudo-exponentiation" title="Schanuel's conjecture">Zilber's pseudo-exponential fields</a> form an AEC.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Common_assumptions">Common assumptions</h2></div>

AECs are very general objects and one usually make some of the assumptions below when studying them:

<ul><li>An AEC has joint embedding if any two model can be embedded inside a common model.</li>

<li>An AEC has no maximal model if any model has a proper extension.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> has amalgamation if for any triple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{0},M_{1},M_{2}\in K}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{0},M_{1},M_{2}\in K}</annotation>

</semantics>

</math><img src="./a1b35d8fbb83c66edd27d0ab3c69df2a62e9ee8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.899ex; height:2.509ex;" alt="{\displaystyle M_{0},M_{1},M_{2}\in K}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{0}\prec _{K}M_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{0}\prec _{K}M_{1}}</annotation>

</semantics>

</math><img src="./e8373944ce8b2361d10bcea88a9e52b8a5da10e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.408ex; height:2.509ex;" alt="{\displaystyle M_{0}\prec _{K}M_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{0}\prec _{K}M_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{0}\prec _{K}M_{2}}</annotation>

</semantics>

</math><img src="./39be26b8c5866bbc90dc168ac97968b7100c80d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.408ex; height:2.509ex;" alt="{\displaystyle M_{0}\prec _{K}M_{2}}" loading="lazy">, there is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\in K}</annotation>

</semantics>

</math><img src="./58179cbdea1d86a9a4870df6c8571f86b0fabf40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle N\in K}" loading="lazy"> and embeddings of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation>

</semantics>

</math><img src="./577d686fc81d1d1eb3ae54e78aeee8957baf6718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation>

</semantics>

</math><img src="./0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}" loading="lazy"> inside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> that fix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{0}}</annotation>

</semantics>

</math><img src="./909a62f3ff41169372143733d3767afe0ad3b14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{0}}" loading="lazy"> pointwise.</li></ul>

Note that in elementary classes, joint embedding holds whenever the theory is <a href="Complete_theory" title="Complete theory">complete</a>, while amalgamation and no maximal models are well-known consequences of the <a href="Compactness_theorem" title="Compactness theorem">compactness theorem</a>. These three assumptions allow us to build a universal model-homogeneous monster model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}}</annotation>

</semantics>

</math><img src="./c8cfb4d0d9067ed01e04e20160d7f644bf4955ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.425ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {C}}}" loading="lazy">, exactly as in the elementary case.

Another assumption that one can make is <a href="Tame_abstract_elementary_class" title="Tame abstract elementary class">tameness</a>.

<div class="mw-heading mw-heading2"><h2 id="Shelah's_categoricity_conjecture">Shelah's categoricity conjecture</h2></div>

Shelah introduced AECs to provide a uniform framework in which to generalize first-order <a href="Stable_theory" title="Stable theory">classification theory</a>. Classification theory started with <a href="Morley's_categoricity_theorem" class="mw-redirect" title="Morley's categoricity theorem">Morley's categoricity theorem</a>, so it is natural to ask whether a similar result holds in AECs. This is Shelah's eventual categoricity conjecture. It states that there should be a Hanf number for categoricity:

For every AEC K there should be a cardinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> depending only on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {LS} (K)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {LS} (K)}</annotation>

</semantics>

</math><img src="./0b762e549c2ad12841c79e364ee9e55c9a289dae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.621ex; height:2.843ex;" alt="{\displaystyle \operatorname {LS} (K)}" loading="lazy"> such that if K is categorical in some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \geq \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \geq \mu }</annotation>

</semantics>

</math><img src="./518ce2853623c4d1e9fcaaabe63d59288ca8fe5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.855ex; height:2.676ex;" alt="{\displaystyle \lambda \geq \mu }" loading="lazy"> (i.e. K has exactly one (up to isomorphism) model of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">), then K is categorical in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \geq \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \geq \mu }</annotation>

</semantics>

</math><img src="./57d7aadaab1bd6d0cd0cb4fb4a35f3392d211a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle \theta \geq \mu }" loading="lazy">.

Shelah also has several stronger conjectures: The threshold cardinal for categoricity is the Hanf number of pseudoelementary classes in a language of cardinality LS(K). More specifically when the class is in a countable language and axiomaziable by an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\omega _{1},\omega }}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }}</annotation>

</semantics>

</math><img src="./037b84dcf7eb85313f964632c939bce6464a657c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.149ex; height:2.843ex;" alt="{\displaystyle L_{\omega _{1},\omega }}" loading="lazy"> sentence the threshold number for categoricity is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{\omega _{1}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beth _{\omega _{1}}}</annotation>

</semantics>

</math><img src="./8083fd95e0b1059007a755af177597c48324a5fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.05ex; width:3.686ex; height:3.009ex;" alt="{\displaystyle \beth _{\omega _{1}}}" loading="lazy">. This conjecture dates back to 1976.

Several approximations have been published (see for example the results section below), assuming <a href="Set_theory" title="Set theory">set-theoretic</a> assumptions (such as the existence of <a href="Large_cardinals" class="mw-redirect" title="Large cardinals">large cardinals</a> or variations of the <a href="Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a>), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.

<div class="mw-heading mw-heading2"><h2 id="Results">Results</h2></div>

The following are some important results about AECs. Except for the last, all results are due to Shelah.

<ul><li>Shelah's Presentation Theorem:<a href="#cite_note-3">[3]</a> Any AEC <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PC} _{2^{\operatorname {LS} (K)}}}">

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {PC} _{2^{\operatorname {LS} (K)}}}</annotation>

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</math><img src="./ed3701b24cbabfd3e327772f17ad95d1ee0e0479.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.28ex; height:2.843ex;" alt="{\displaystyle \operatorname {PC} _{2^{\operatorname {LS} (K)}}}" loading="lazy">: it is a reduct of a class of models of a first-order theory omitting at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\operatorname {LS} (K)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{\operatorname {LS} (K)}}</annotation>

</semantics>

</math><img src="./a4b669a7a8e1b0a59b5a2588655c20a20c5d5567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.076ex; height:2.843ex;" alt="{\displaystyle 2^{\operatorname {LS} (K)}}" loading="lazy"> <a href="Type_(model_theory)" title="Type (model theory)">types</a>.</li>

<li>Hanf number for existence:<a href="#cite_note-4">[4]</a> Any AEC <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> which has a model of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{(2^{\operatorname {LS} (K)})^{+}}}">

<msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beth _{(2^{\operatorname {LS} (K)})^{+}}}</annotation>

</semantics>

</math><img src="./bcf43dbee440a79cfe9ee12fdd14ce82f83b9166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; margin-left: -0.05ex; width:9.101ex; height:3.509ex;" alt="{\displaystyle \beth _{(2^{\operatorname {LS} (K)})^{+}}}" loading="lazy"> has models of arbitrarily large sizes.</li>

<li>Amalgamation from categoricity:<a href="#cite_note-5">[5]</a> If K is an AEC categorical in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda ^{+}}</annotation>

</semantics>

</math><img src="./2c84f0723364f838bbe469a1fc2fb52ede4d844a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.866ex; height:2.509ex;" alt="{\displaystyle \lambda ^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\lambda }<2^{\lambda ^{+}}}">

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{\lambda }<2^{\lambda ^{+}}}</annotation>

</semantics>

</math><img src="./1ab3b333e36892d33fda8f9c119b04d375113fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.007ex; height:3.009ex;" alt="{\displaystyle 2^{\lambda }<2^{\lambda ^{+}}}" loading="lazy">, then K has amalgamation for models of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">.</li>

<li>Existence from categoricity:<a href="#cite_note-6">[6]</a> If K is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PC} _{\aleph _{0}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {PC} _{\aleph _{0}}}</annotation>

</semantics>

</math><img src="./5b88200534a8e8e019d8017af8fd45a1e2deb0f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.329ex; height:2.843ex;" alt="{\displaystyle \operatorname {PC} _{\aleph _{0}}}" loading="lazy"> AEC with Löwenheim–Skolem number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation>

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</math><img src="./721cd7f8c15a2e72ad162bdfa5baea8eef98aab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}" loading="lazy"> and K is categorical in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation>

</semantics>

</math><img src="./721cd7f8c15a2e72ad162bdfa5baea8eef98aab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation>

</semantics>

</math><img src="./78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}" loading="lazy">, then K has a model of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}}</annotation>

</semantics>

</math><img src="./765bb468708b2eec66bf2dc6505a4c92959d697d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{2}}" loading="lazy">. In particular, no sentence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\omega _{1},\omega }(Q)}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }(Q)}</annotation>

</semantics>

</math><img src="./3d448f1a6cdca64d74c8beeba71dd8dd5f6f92eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.796ex; height:3.009ex;" alt="{\displaystyle L_{\omega _{1},\omega }(Q)}" loading="lazy"> can have exactly one uncountable model.</li>

<li>Approximations to Shelah's categoricity conjecture:

<ul><li>Downward transfer from a successor:<a href="#cite_note-7">[7]</a> If K is an abstract elementary class with amalgamation that is categorical in a "high-enough" <a href="Successor_cardinal" title="Successor cardinal">successor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">, then K is categorical in all high-enough <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \leq \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu \leq \lambda }</annotation>

</semantics>

</math><img src="./df336515508cd87cf8852d124e0b4e220edb55cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.855ex; height:2.676ex;" alt="{\displaystyle \mu \leq \lambda }" loading="lazy">.</li>

<li>Shelah's categoricity conjecture for a successor from large cardinals:<a href="#cite_note-8">[8]</a> If there are class-many <a href="Strongly_compact_cardinal" title="Strongly compact cardinal">strongly compact cardinals</a>, then Shelah's categoricity conjecture holds when we start with categoricity at a successor.</li></ul></li></ul>

<ul><li><a href="Tame_abstract_elementary_class" title="Tame abstract elementary class">Tame abstract elementary class</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFShelah1987">Shelah 1987</a>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFGrossberg2002">Grossberg 2002</a>, Section 1.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFGrossberg2002">Grossberg 2002</a>, Theorem 3.4.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFGrossberg2002">Grossberg 2002</a>, Corollary 3.5. Note that there is a typo there and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{\operatorname {LS} (K)}}}">

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{2^{\operatorname {LS} (K)}}}</annotation>

</semantics>

</math><img src="./246eb3822c2a8060e15dc9eb014494cdec451c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.182ex; height:3.176ex;" alt="{\displaystyle 2^{2^{\operatorname {LS} (K)}}}" loading="lazy"> should be replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\operatorname {LS} (K)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{\operatorname {LS} (K)}}</annotation>

</semantics>

</math><img src="./a4b669a7a8e1b0a59b5a2588655c20a20c5d5567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.076ex; height:2.843ex;" alt="{\displaystyle 2^{\operatorname {LS} (K)}}" loading="lazy">.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFGrossberg2002">Grossberg 2002</a>, Theorem 4.3.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFGrossberg2002">Grossberg 2002</a>, Theorem 5.1.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFShelah1999">Shelah 1999</a>.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> This is due to Will Boney, but combines results of many people, including Grossberg, Makkai, Shelah, and VanDieren. A proof appears in <a href="#CITEREFBoney2014">Boney 2014</a>, Theorem 7.5.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFShelah1987" class="citation cs2"><a href="Saharon_Shelah" title="Saharon Shelah">Shelah, Saharon</a> (1987), John T. Baldwin (ed.), Classification of Non Elementary Classes II. Abstract Elementary Classes, Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, pp. 419–497</cite></li>

<li><cite id="CITEREFShelah1999" class="citation cs2"><a href="Saharon_Shelah" title="Saharon Shelah">Shelah, Saharon</a> (1999), <a rel="nofollow" class="external text" href="http://shelah.logic.at/files/394.pdf">"Categoricity for abstract classes with amalgamation"</a> (PDF), Annals of Pure and Applied Logic, 98 (1): 261–294, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9809197">math/9809197</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0168-0072%2898%2900016-5">10.1016/s0168-0072(98)00016-5</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:27872122">27872122</a></cite></li>

<li><cite id="CITEREFGrossberg2002" class="citation cs2">Grossberg, Rami (2002), <a rel="nofollow" class="external text" href="https://www.math.cmu.edu/~rami/Rami-NBilgi.pdf">"Classification theory for abstract elementary classes"</a> (PDF), Logic and algebra, Contemporary Mathematics, vol. 302, Providence, RI: American Mathematical Society, pp. 165–204, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.6.9630">10.1.1.6.9630</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F302%2F05080">10.1090/conm/302/05080</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780821829844</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1928390">1928390</a></cite></li>

<li><cite id="CITEREFBaldwin,_John_T.2006" class="citation cs2">Baldwin, John T. (July 7, 2006), <a rel="nofollow" class="external text" href="http://www2.math.uic.edu/~jbaldwin/pub/turino2.pdf">Abstract Elementary Classes: Some Answers, More Questions</a> (PDF)</cite></li>

<li><cite id="CITEREFShelah2009" class="citation cs2"><a href="Saharon_Shelah" title="Saharon Shelah">Shelah, Saharon</a> (2009), Classification theory for elementary abstract classes, Studies in Logic (London), vol. 18, College Publications, London, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-904987-71-0</bdi></cite></li>

<li><cite id="CITEREFShelah2009" class="citation cs2"><a href="Saharon_Shelah" title="Saharon Shelah">Shelah, Saharon</a> (2009), Classification theory for abstract elementary classes. Vol. 2, Studies in Logic (London), vol. 20, College Publications, London, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-904987-72-7</bdi></cite></li>

<li><cite id="CITEREFBaldwin2009" class="citation cs2">Baldwin, John T. (2009), <a rel="nofollow" class="external text" href="http://homepages.math.uic.edu/~jbaldwin/model11.html">Categoricity</a>, University Lecture Series, vol. 50, American Mathematical Society, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0821848937</bdi></cite></li>

<li><cite id="CITEREFBoney2014" class="citation arxiv cs1">Boney, Will (2014). "Tameness from large cardinal axioms". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1303.0550v4">1303.0550v4</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite></li></ul>

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