# nLab maximal consistent

Given a normal modal logic, $\Lambda$, a set, $\Gamma$ of formulae is said to be $\Lambda$-consistent if $¬\left(\Gamma {⊢}_{\Lambda }\perp \right)$, i.e., $\perp$ is not deducible from $\Gamma$.

A set, $\Gamma$, of formulae is said to be $\Lambda$-maximal if it is consistent and, for any $\varphi \in {ℒ}_{\omega }\left(n\right)$ either $\varphi \in \Gamma$ or $¬\varphi \in \Gamma$.

#### Important

If $\Gamma$ is a $\Lambda$-maximal set of formulae, then within the Lindenbaum-Tarski algebra, ${𝔄}_{\omega }^{\Lambda }$, the set ${x}_{\Lambda }=\left\{\mid \mid \varphi \mid \mid \mid \varphi \in \Lambda \right\}$ is an ultrafilter.

## Canonical frame

Let ${S}_{\omega }^{\Lambda }=\left\{\Gamma \mid \Gamma \mathrm{is}\Lambda -\mathrm{maximal}\right\}$, then $\Gamma ↔{x}_{\Gamma }$ is a bijection between ${S}_{\omega }^{\Lambda }$ and the set of ultrafilters of ${𝔄}_{\omega }^{\Lambda }$.

This set forms the set of states / worlds for the canonical frame of $\Lambda$. The relations are given by

${R}_{i}\Gamma \Delta \mathrm{if},\mathrm{and}\mathrm{only}\mathrm{if},{\diamond }_{i}\Delta \subseteq \Gamma .$R_i \Gamma\Delta if, and only if, \Diamond_i\Delta \subseteq \Gamma.
Revised on November 5, 2010 07:59:06 by Tim Porter (95.147.238.17)