K-inconsistency

A sequence of formulas $$\phi_i(x;a_i)$$ is said to be $$k$$-inconsistent if for every $$\{i_1,\ldots,i_k\}$$ of size $$k$$, $$\bigwedge_{j = 1}^k \phi_{i_j}(x;a_{i_j})$$ is inconsistent. That is, a sequence of formulas is $$k$$-inconsistent if any $$k$$ of the formulas in the sequence is jointly inconsistent. For example, 2-inconsistency is equivalent to pairwise inconsistency.

Typically, $$k$$-inconsistency is only considered when the $$\phi_i(x;y)$$ are all the same formula.

This notion is rigged to behave very well with respect to indiscernible sequences. Specifically:


 * If $$b_1, b_2, \ldots$$ is an indiscernible sequence, then $$\{\phi(x;b_i)\}$$ is inconsistent if and only if it is $$k$$-inconsistent for some $$k$$.
 * If $$b_1, b_2, \ldots$$ is arbitrary, and $$\{\phi(x;b_i)\}$$ is $$k$$-inconsistent, then this is witnessed in the EM-type of $$\langle b_i \rangle_i$$. Consequently, if $$c_1, c_2, \ldots$$ is an indiscernible sequence extracted from $$b_1, b_2, \ldots$$, then $$\{\phi(x;c_i)\}$$ will also be $$k$$-inconsistent, for the same $$k$$.

$$k$$-inconsistency plays a basic role in the definitions of dividing, forking, and their variants (such as thorn-forking).