Editing Consistency (section)

===Henkin's theorem===
Let <math>S</math> be a [[signature (logic)|set of symbols]]. Let <math>\Phi</math> be a maximally consistent set of <math>S</math>-formulas containing [[Witness (mathematics)#Henkin witnesses|witnesses]].

Define an [[equivalence relation]] <math>\sim</math> on the set of <math>S</math>-terms by <math>t_0 \sim t_1</math> if <math>\; t_0 \equiv t_1 \in \Phi</math>, where <math>\equiv</math> denotes [[First-order logic#Equality and its axioms|equality]]. Let <math>\overline t</math> denote the [[equivalence class]] of terms containing <math>t </math>; and let <math>T_\Phi := \{ \; \overline t \mid t \in T^S \} </math> where <math>T^S </math> is the set of terms based on the set of symbols <math>S</math>.

Define the <math>S</math>-[[Structure (mathematical logic)|structure]] <math>\mathfrak T_\Phi </math> over <math> T_\Phi </math>, also called the '''term-structure''' corresponding to <math>\Phi</math>, by:

# for each <math>n</math>-ary relation symbol <math>R \in S</math>, define <math>R^{\mathfrak T_\Phi} \overline {t_0} \ldots \overline {t_{n-1}}</math> if <math>\; R t_0 \ldots t_{n-1} \in \Phi;</math><ref>This definition is independent of the choice of <math>t_i</math> due to the substitutivity properties of <math>\equiv</math> and the maximal consistency of <math>\Phi</math>.</ref>
# for each <math>n</math>-ary function symbol <math>f \in S</math>, define <math>f^{\mathfrak T_\Phi} (\overline {t_0} \ldots \overline {t_{n-1}}) := \overline {f t_0 \ldots t_{n-1}};</math>
# for each constant symbol <math>c \in S</math>, define <math>c^{\mathfrak T_\Phi}:= \overline c.</math>

Define a variable assignment <math>\beta_\Phi</math> by <math>\beta_\Phi (x) := \bar x</math> for each variable <math>x</math>. Let <math>\mathfrak I_\Phi := (\mathfrak T_\Phi,\beta_\Phi)</math> be the '''term [[Interpretation (logic)#First-order logic|interpretation]]''' associated with <math>\Phi</math>.

Then for each <math>S</math>-formula <math>\varphi</math>:

{{center|1=
<math>\mathfrak I_\Phi \vDash \varphi</math> if and only if <math> \; \varphi \in \Phi.</math>{{citation needed|date=September 2018}}
}}