Shelah’s categoricity conjecture from a successor for tame abstract elementary classes

Grossberg, Rami; VanDieren, Monica

J. Symbolic Logic, Tome 71 (2006) no. 1, p. 553-568 / Harvested from Project Euclid

Résumé

We prove a categoricity transfer theorem for tame abstract elementary classes. ¶ Theorem. Suppose that 𝔎 is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ≥Max{χ,LS(𝔎)⁺}. If 𝔎 is categorical in λ and λ⁺, then 𝔎 is categorical in λ⁺⁺. ¶ Combining this theorem with some results from [37], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes: ¶ Corollary. Suppose 𝔎 is a χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ₀:= Hanf(𝔎). If χ≤ℶ_(2^μ₀)⁺ and 𝔎 is categorical in some λ⁺>ℶ_(2^μ₀)⁺, then 𝔎 is categorical in μ for all μ>ℶ_(2^μ₀)⁺.

Publié le : 2006-06-14
Classification: 03C45, 03C52, 03C75, 03C05, 03C55, 03C95

@article{1146620158,
     author = {Grossberg, Rami and VanDieren, Monica},
     title = {Shelah's categoricity conjecture from a successor for tame abstract elementary classes},
     journal = {J. Symbolic Logic},
     volume = {71},
     number = {1},
     year = {2006},
     pages = { 553-568},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1146620158}
}

Grossberg, Rami; VanDieren, Monica. Shelah’s categoricity conjecture from a successor for tame abstract elementary classes. J. Symbolic Logic, Tome 71 (2006) no. 1, pp.  553-568. http://gdmltest.u-ga.fr/item/1146620158/