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). ::: ::: :::