indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
In model theory, given a cardinal $\kappa$, a theory is $\kappa$-categorical (or categorical in cardinality $\kappa$), if it has precisely one isomorphism class of models of cardinality $\kappa$.
The Morley categoricity theorem says that a first-order theory $T$ with countably many symbols is $\kappa$-categorical for one uncountable cardinal $\kappa$ iff $T$ is categorical in any uncountable cardinality.
The main interest is beyond first order. There is much work there, for non-elementary classes, centered around (various versions of) Shelah’s Categoricity Conjecture especially in the setting of abstract elementary classes. Also, very interesting program is initiated by Zilber in early 2000s.
Study of categoricity lead historically to the development of the stability theory in model theory, see also geometric stability theory.
John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf
John T. Baldwin, What is a complete theory, talk, pdf
Morley’s categoricity theorem is from
M. Morley, Categoricity in power, Trans. Amer. Math. Soc. 114:514–538, 1965.
wikipedia: Morley categoricity theorem
Saharon Shelah, Classification theory and the number of non-isomorphic models, 2nd ed. North Holland Amsterdam, 1990.
Saharon Shelah, Categoricity of abstract classes with amalgamation, Annals of Pure and Applied Logic 98(1-3) 241–294, 1999.
Boris Zilber, Uncountably categorical theories, Transl. Math. Monog., Amer. Math. Soc. 1993
Rami Grossberg, Monica Vandieren, Shelah’s categoricity conjecture from a successor for tame AECs, pdf
Michael Makkai, Saharon Shelah, Categoricity of theories $L_{\kappa\omega}$ with $\kappa$ a compact cardinal, Annals of Pure and Applied Logic 47, 41-97, 1990
Christian Espíndola, An extension of Shelah’s eventual categoricity conjecture to accessible categories with directed colimits, arxiv/1906.09169; A short proof of Shelah’s eventual categoricity conjecture for AEC’s with interpolation, under GCH, arxiv/1909.13713; A topos-theoretic proof of Shelah’s eventual categoricity conjecture, Brno, April 30, 2020, video yt, slides pdf
Adam Harris, Categoricity and covering spaces, PhD thesis, Merton college arXiv:1412.3484
C. Daw, A. Harris, Categoricity of modular and Shimura curves, J. Inst. Math. Jussieu, 16(5) (2017) 1075–1101 doi
Boris Zilber, Chris Daw, Modular curves and their pseudo-analytic cover, arXiv:2107.11110; Canonical models of modular curves and the Galois action on CM-points, arXiv:2106.06387
Last revised on October 7, 2022 at 18:06:14. See the history of this page for a list of all contributions to it.