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).