4.The definition of canonical structure $\mathbf{\alpha}$ for
$\mathbf{T}$
$\left\{
|α|={a°
1
,...,a
°
n
,...}
f
α
(a
°
1
,...,a
°
n
)=(fa
1
...a
n
)
°
p
α
(a
°
1
,...,a
°
n
)iff?
T
pa
1
...a
n
\right. $
where,the equivalence class of $\mathbf{a}$ is designated by
$\mathbf{a^{\circ}}$,i.e.,all the equivalence $\mathbf{a_1,b_1,...}$ is
designated by $\mathbf{a^{\circ}}$, that is,
$\mathbf{a_1^{\circ} \sim a_1 ,..., a_n^{\circ} \sim a_n}$
$\mathbf{a_1^{\circ} \sim b_1 ,..., a_n^{\circ} \sim b_n}$
...
We define $\mathbf{a \sim b}$ to mean $\mathbf{\vdash_{T}a=b}$.Then
$\mathbf{a \sim b}$ is an equivalence relation,that is,
$ \left\{
a～aa～b→(a～c?b～c)
a～b→b～a
\right. $
Proof.
By the identity axioms, $\mathbf{\vdash_{T}a=a}$, we get $\mathbf{a \sim
a}$.
By the equality theorem, let $\mathbf{A‘}$ be obtained from $\mathbf{A}$
which means $\mathbf{a=c}$ by replacing $\mathbf{a}$ by $\mathbf{b}$. So if
$\mathbf{\vdash_{T}a=b}$, then $\mathbf{\vdash_{T}A \leftrightarrow A‘}$, i.e.,
$\mathbf{\vdash_{T}a=c \leftrightarrow b=c}$. $\mathbf{a=b \rightarrow (a=c
\leftrightarrow b=c)}$ is tautological consequence of $\mathbf{a=c
\leftrightarrow b=c}$, and $\mathbf{\vdash_{T}a=c \leftrightarrow b=c}$, then
$\mathbf{\vdash_{T}a=b \rightarrow (a=c \leftrightarrow b=c)}$ by tautology
theorem. That is,$\mathbf{a \sim b \rightarrow (a \sim c \leftrightarrow b \sim
c )}$
$\mathbf{\vdash_{T}a=b \rightarrow (a=c \leftrightarrow b=c)}\\ \Rightarrow
\mathbf{\vdash_{T}a=b \rightarrow (a=a \leftrightarrow b=a)}\\ \Rightarrow
\mathbf{\vdash_{T[a=b]} a=a \leftrightarrow b=a} \ \text{by deduction theorem}\\
\Rightarrow \mathbf{\vdash_{T[a=b]} b=c \quad \text{iff} \quad \vdash_{T[a=b]}
a=a} \ \text{by tautology theorem}\\ \Rightarrow \mathbf{\vdash_{T} a=b
\leftrightarrow b=c} \ \text{by deduction theorem}$
That is, $\mathbf{a \sim b \rightarrow b \sim a}$
4.2 THE COMPLETENESS THEOREM: (4) The definition of canonical
structure $\mathbf{\alpha}$ for $\mathbf{T}$