indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
In model theory the notion of abstract elementary classes is a vast generalizations of that of elementary classes of structures beyond first-order theories (e.g. for the infinitary logic $L_{\omega_1,\omega}$) as introduced by Saharon Shelah. Their theory is also more general than the homogeneous model theory.
An abstract elementary class is a nonempty class $K$ of structures for a given signature with language $L(K)$, that is closed under isomorphisms and equipped with a strong substructure relation $\prec_K$ (strong substructure relation means that if $M\prec_K N$ and $M_0\subset M$ is a substructure, then $M_0 \prec_K N$) that is a partial order satisfying the axioms on union of chains (Tarski-Vaught), coherence and downward Loewenheim-Skolem properties. More precisely, $\prec_K$ is a partial order such that
(A0) if $M,N\in K$, $M\prec_K N$ then $M\subset N$
(A1) (closure under isomorphisms)
(a) $M\in K$ and $N$ an $L(K)$ structure with $N\cong M$, then $N\in K$
(b) if $N_1,N_2,M_1,M_2\in K$, $f_i : N_i\cong M_i$, $i = 1,2$, $f_1\subset f_2$, with $M_1\prec_K M_2$ then $N_1\prec_K N_2$
(A2) for $M,N,P\in K$, if $M\prec_K P$, $N\prec_K P$, and $M\subset N$, then $M\prec_K N$
(A3) downward Loewenheim-Skolem. There exist a cardinal $LS(K) = LS(K,\prec_K)\geq |L(K)|+\aleph_0$ such that $\forall M\in K$, $\forall A\subset |M|$, $\exists N\in K$ with $A\subset |N|$, $N\prec_K M$, $\|N\|\leq |A|+LS(K)$.
(A4) (Tarski-Vaught chain condition) for every regular cardinal $\mu$
….
The usual elementary classes, i.e. the classes of the form $K = Mod(T)$ for a first-order theory $T$, are abstract elementary with respect to the relation $\prec_K$ of being an elementary submodel, with ${|LS(K)|} = {|L(T)|}+\aleph_0$ ($L(T)$ is the underlying language of the theory $T$).
wikipedia abstract elementary class
Saharon Shelah, Classification theory for elementary abstract classes I, II, Studies in Logic (London), 18, 20, College Publications, London 2009
John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf
D. W. Kueker, Abstract elementary classes and infinitary logic, Ann. Pure Appl. Logic 156 (2008), 274-286.
AECs can also be essentially identified with accessible categories in which all morphisms are monomorphisms. Some recent papers which study them from this viewpoint include:
Tibor Beke, Jiří Rosický Abstract elementary classes and accessible categories, Annals of Pure and Applied Logic 163 (2012) 2008-2017, arxiv/1005.2910
Michael Lieberman, Jiří Rosický, Sebastien Vasey, Internal sizes in μ-abstract elementary classes, arxiv/1708.06782; Set-theoretic aspects of accessible categories, arxiv/1902.06777
M. J. Lieberman, Topological and category-theoretic aspects of abstract elementary classes, Thesis, The University of Michigan 2009, pdf; defense slides pdf; Category theoretic aspects of abstract elementary classes, Annals Pure Appl. Logic 162 (2011), 903-915; A topology for Galois types in AECs, arxiv/0906.3573
On topos theoretic methods in model theory of AECs:
On the homotopy types of classifying spaces of abstract elementary classes:
225 5 (2021) 106461 [arXiv:1909.07965, doi:10.1016/j.jpaa.2020.106461]
Last revised on March 2, 2023 at 16:52:59. See the history of this page for a list of all contributions to it.