FANDOM


Fix some theory $ T $. Let $ \kappa $ be a cardinal. An inp pattern of depth $ \kappa $ is a collection of formulas $ \langle \phi_\alpha(x;y) \rangle_{\alpha < \kappa} $ and constants $ b_{\alpha,i} $ for $ \alpha < \kappa $ and $ i < \omega $ and integers $ k_\alpha < \omega $ such that for every $ \alpha < \kappa $, the set of formulas $ \{\phi(x;b_{\alpha,i}) : i < \omega\} $ is $ k_\alpha $-inconsistent, but for every function $ \eta : \kappa \to \omega $, the collection $ \{\phi(x;b_{\alpha,\eta(\alpha)}) : \alpha < \kappa\} $ is consistent.

More generally, if $ \Sigma(x) $ is a partial type, an inp pattern of depth $ \kappa $ in $ \Sigma(x) $ is an inp pattern of depth $ \kappa $ such that for every $ \eta : \kappa \to \omega $, $ \Sigma(x) \cup \{\phi(x;b_{\alpha,\eta(\alpha)}) : \alpha < \kappa\} $ is consistent.

Shelah defines $ \kappa_{inp} $ of a theory to be the supremum of the depths of possible inp-patterns. Hans Adler (right?) defines the burden of a partial type $ \Sigma(x) $ to be the supremum of the depths of the inp patterns in $ \Sigma(x) $. A theory is said to be strong if there are no inp patterns of depth $ \omega $. A theory is $ NTP_2 $ if and only if $ \kappa_{inp} < \infty $.

Artem Chernikov (right?) proved that burden is submultiplicative in the following sense: if $ bdn(b/C) < \kappa $ and $ bdn(a/bC) < \lambda $, then $ bdn(ab/C) < \kappa \times \lambda $. It is conjectured that burden is subadditive ($ bdn(ab/C) \le bdn(a/bC) + bdn(b/C) $), but this is unknown.

Given an inp pattern of depth $ \kappa $, one can always find an inp pattern of the same depth, using the same formulas and same $ k_\alpha $'s, such that the rows $ \langle b_{\alpha,i}\rangle_{i < \omega} $ are mutually indiscernible. Given mutual indiscernibility, the $ k_\alpha $-inconsistence can be rephrased as inconsistency. And the only vertical path one must check is the leftmost column. So one may also define the burden of $ \Sigma(x) $ to be the supremum of the $ \kappa $ for which there exists $ \kappa $ mutually indiscernible sequences $ \langle b_{\alpha,i} \rangle_i $ for $ \alpha < \kappa $ and formulas $ \phi_\alpha(x;y) $ for $ \alpha < \kappa $ such that for each $ \alpha $, $ \{\phi_\alpha(x;b_{\alpha,i}) : i < \omega\} $ is inconsistent, and $ \Sigma(x) \cup \{\phi_\alpha(x;b_{\alpha,0}) : \alpha < \kappa\} $ is consistent.

Relation to ict patterns Edit

Any mutually indiscernible inp pattern is already a mutually indiscernible ict pattern. Under the hypothesis of NIP, a mutually indiscernible ict pattern of depth $ \kappa $ can be converted to a mutually indiscernible inp pattern of the same depth, as follows. If the original ict pattern is $ \{\phi_\alpha(x;b_{\alpha,i}) $, then we take as our inp pattern the array of formulas whose entry in the $ \alpha $th row and $ i $th column is $ \phi_\alpha(x;b_{\alpha,2i}) \wedge \neg \phi_\alpha(x;b_{\alpha,2i+1}) $. The "no alternation" characterization of NIP implies that each row is inconsistent. The ict condition ensures that we can find an $ a $ satisfying $ \phi_\alpha(x;b_{\alpha,0}) $ and $ \neg \phi_\alpha(x;b_{\alpha,1}) $ for every $ \alpha $, showing that the first column is consistent.

Consequently, if NIP holds (equivalently, $ \kappa_{ict} < \infty $), then $ \kappa_{inp} = \kappa_{ict} $, and the burden of any type equals its dp-rank. Also, a theory is strongly dependent if and only if it is strong and NIP (dependent).