4.2 THE COMPLETENESS THEOREM: (1) The second form of the completeness theorem implies the first.

4.2 THE COMPLETENESS THEOREM

Completeness Theorem,First Form (G${\ddot{o}}$del). A formula $\mathbf{A}$ of a theory $\mathbf{T}$ is a theorem of $\mathbf{T}$ iff it is valid in $\mathbf{T}$.

Completeness Theorem,Second Form. A theory $\mathbf{T}$ is consistent iff it has a model

 

(1) The second form of the completeness theorem implies the first.

Proof

From second form, we get

\begin{equation}\label{SecondForm} \mathbf{T[\neg A']} \ \text{is consistent} \Leftrightarrow \mathbf{T[\neg A']} \ \text{has a model} \ \end{equation}

where $\mathbf{A}$ is theorem of $\mathbf{T}$, $\mathbf{A'}$ is closure of $\mathbf{A}$,

that is

\[\mathbf{T[\neg A']} \ \text{is inconsistent} \Leftrightarrow \mathbf{T[\neg A']} \ \text{has no model} \ \]

The left of \eqref{SecondForm}, by the corollary to reduction theorem for consistency(P43), we get

\[\mathbf{T[\neg A']} \ \text{is inconsistent} \Leftrightarrow \mathbf{\vdash_T A} \]

The right of \eqref{SecondForm}, $\mathbf{\neg A'}$ is a nonlogical axiom in theory $\mathbf{T[\neg A']}$ by the definition of $\mathbf{T[\Gamma]}$(P42). So, by the definition of model(P22), i.e., all the nonlogical axioms of theory are valid, we get

\[\mathbf{T[\neg A']} \ \text{has a model} \Leftrightarrow \mathbf{\neg A'} \ \text{is valid} \]

that is

\[\mathbf{T[\neg A']} \ \text{has no model} \Leftrightarrow \mathbf{\neg A'} \ \text{is not valid} \]

\[\mathbf{\neg A'} \ \text{is not valid} \Leftrightarrow \mathbf{A'} \ \text{is valid} \]

By the corollary of closure theory (P32),we get \[\mathbf{A'} \ \text{is valid} \Leftrightarrow \mathbf{A} \ \text{is valid} \]

i.e., \[\mathbf{T[\neg A']} \ \text{has no model} \Leftrightarrow \mathbf{A} \ \text{is valid} \]

thus, we get the first form \begin{equation} \mathbf{\vdash_T A} \Leftrightarrow \mathbf{A} \ \text{is valid} \end{equation}