On categoricity in successive cardinals

This is the title of a talk given at the Harvard Logic seminar on February 4, 2019.

Abstract

The upward Löwenheim-Skolem theorem says that any first-order theory with an infinite model has arbitrarily large models. Such a result completely completely breaks down when we start looking at infinitary logics. In this case, one can still ask what properties of small models imply the existence of larger models.

One of the simplest such property is categoricity (the existence of a unique model of a fixed cardinality). In that direction, Shelah proved the remarkable result that an L_{ω1, ω} sentence categorical in ω₀ and ω₁ has a model of cardinality ω₂. Generalizations of this result have been key motivating questions for the development of a classification theory of infinitary logics, and also have also yielded some interesting set theory. We will survey some old results as well as recent developments.

References

• Sebastien Vasey, On categoricity in successive cardinals, Preprint: pdf arXiv, 18 pages.

• Marcos Mazari-Armida and Sebastien Vasey, Universal classes near ℵ₁, The Journal of Symbolic Logic 83 (2018), no. 4, 1633–1643.

• Rami Grossberg. Classification theory for abstract elementary classes, Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics, Vol 302, AMS, (2002), pp. 165-204. pdf
• Saharon Shelah. The number of uncountable models of Ψ ∈ L_ω1,ω. Part A, Israel Journal of Mathematics 46 (1983), no. 3, 212-240.
• Saharon Shelah. The number of uncountable models of Ψ ∈ L_ω1,ω. Part B, Israel Journal of Mathematics 46 (1983), no. 4, 241-273.

• John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.

• Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vols. 18 and 20, College Publications, 2009.