## FANDOM

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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).