279 research outputs found
On Hanf numbers of the infinitary order property
We study several cardinal, and ordinal--valued functions that are relatives
of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq
L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an
ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min
{mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the
(phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+,
omega}, such that for every chi >= kappa, T has the (phi', chi)-order
property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in
L_{kappa^+,omega}}
Model theoretic stability and definability of types, after A. Grothendieck
We point out how the "Fundamental Theorem of Stability Theory", namely the
equivalence between the "non order property" and definability of types, proved
by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's
"Crit{\`e}res de compacit{\'e}" from 1952. The familiar forms for the defining
formulae then follow using Mazur's Lemma regarding weak convergence in Banach
spaces
Upward Stability Transfer for Tame Abstract Elementary Classes
Grossberg and VanDieren have started a program to develop a stability theory
for tame classes. We prove, for instance, that for tame abstract elementary
classes satisfying the amlagamation property and for large enough cardinals
kappa, stability in kappa implies stability in kappa^{+n} for each natural
number n
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