Editing Consistency (section)

===Basic results===
# The following are equivalent:
## <math>\operatorname{Inc}\Phi</math>
## For all <math>\varphi,\; \Phi \vdash \varphi.</math>
# Every satisfiable set of formulas is consistent, where a set of formulas <math>\Phi</math> is satisfiable if and only if there exists a model <math>\mathfrak{I}</math> such that <math>\mathfrak{I} \vDash \Phi </math>.
# For all <math>\Phi</math> and <math>\varphi</math>:
## if not <math> \Phi \vdash \varphi</math>, then <math>\operatorname{Con}\left( \Phi \cup \{\lnot\varphi\}\right)</math>;
## if <math>\operatorname{Con}\Phi</math> and <math>\Phi \vdash \varphi</math>, then <math> \operatorname{Con} \left(\Phi \cup \{\varphi\}\right)</math>;
## if <math>\operatorname{Con}\Phi</math>, then <math>\operatorname{Con}\left( \Phi \cup \{\varphi\}\right)</math> or <math>\operatorname{Con}\left( \Phi \cup \{\lnot \varphi\}\right)</math>.
# Let <math>\Phi</math> be a maximally consistent set of formulas and suppose it contains [[Witness (mathematics)|witnesses]].  For all <math>\varphi</math> and <math> \psi </math>:
## if <math> \Phi \vdash \varphi</math>, then <math>\varphi \in \Phi</math>,
## either <math>\varphi \in \Phi</math> or <math>\lnot \varphi \in \Phi</math>,
## <math>(\varphi \lor \psi) \in \Phi</math> if and only if <math>\varphi \in \Phi</math> or <math>\psi \in \Phi</math>,
## if <math>(\varphi\to\psi) \in \Phi</math> and <math>\varphi \in \Phi </math>, then <math>\psi \in \Phi</math>,
## <math>\exists x \, \varphi \in \Phi</math> if and only if there is a term <math>t</math> such that <math>\varphi{t \over x}\in\Phi</math>.{{citation needed|date=September 2018}}