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  In the first case, we halt and say we have a
  member of $T^*$. In the second case, we check if
  $A = f(n)$, saying we have a member of $T^*$ if
  so, and that we don't otherwise.

  We can do this because we can scan the list
  $\{f(n) : n < \omega\}$ and check symbol by
  symbol whether $f(n)$ matches $A$, which takes
  finite time.

  If the input is not of the right shape, we halt
  and decide that it is $\notin T^*$.
\end{proof}

\begin{fclemma}
  \label{lemma:2.2.5}
  % Lemma 2.2.5
  The set of (\glspl{Gn} for) \gls{tr} functions
  is not \gls{renum}.
\end{fclemma}

\begin{proof}
  Suppose otherwise, so there is a \gls{tr}
  function whose image is the set of \glspl{Gn} of
  \gls{tr} functions.

  So for any \gls{tr} $r$, there is $n$ such that
  $\Gn{f(n)} = r$. Define $g
  : \Nbb \to \Nbb$ by $g(n) = \Gn{f(n)}(n) + 1$.
  This is certainly \gls{tr}, but can't be the
  function coded by $f(m)$ for any $m$,
  contradiction.
\end{proof}

\begin{fcdefn}[Language of arithmetic]
  \glssymboldefn{lpa}%
  \glsnoundefn{bta}{base theory of arithmetic}{NA}%
  \glsnoundefn{PA}{PA}{NA}%
  \label{defn:2.2.6}%
  % Definition 2.2.6
  The language of arithmetic is the first-order
  language $L_{\text{PA}}$ with signature $(0, 1,
  +, \cdot, <)$. The \emph{base theory of
  arithmetic} is the $L_{\text{PA}}$-theory
  $\text{PA}^-$ whose
  axioms express that:
  \begin{cenum}[(1)]
    \item $+$ and $\cdot$ are commutative and
      associative, with identity elements $0$ and
      $1$ respectively;
    \item $\cdot$ distributes over $+$;
    \item $<$ is a linear ordering compatible with
      $+$ and $\cdot$;
    \item $\forall x . \forall y . (x < y \to
      \exists z . x + z = y)$;
    \item $0 < 1 \wedge \forall x . (x > 0 \to x
      \ge 1)$;
    \item $\forall x . x \ge 0$.
  \end{cenum}
  The (first-order) theory of Peano arithmetic
  PA is obtained from $\text{PA}^-$ by adding the
  \emph{scheme of induction}: for each
  $L_{\text{PA}}$-formula $\varphi(x, \ol{y})$,
  the axiom
  \[
    I\varphi \defeq \forall \ol{y} . (\varphi(0,
    \ol{y}) \wedge \forall x . (\varphi(x, \ol{y})
    \to \varphi(x + 1, \ol{y})) \to \forall x .
    \varphi(x, \ol{y}))
  .\]
\end{fcdefn}

\begin{fcdefn}[Delta0-formula, Sigma1-formula]
  \glssymboldefn{delz}%
  \label{defn:2.2.7}
  % Definition 2.2.7
  A \emph{$\Delta_0$-formula} of \gls{PA} is one
  whose quantifiers are bounded, i.e. $\exists x <
  t . \varphi(x)$ or $\forall x < t . \varphi(x)$,
  where $t$ is not free in $\varphi$ and $\varphi$
  is quantifier free.

  We say $\varphi(\ol{x})$ is a $\Sigma_1$-formula
  if there is a $\Delta_0$-formula $\psi(\ol{x},
  \ol{y})$ such that
  \[
    \PA \syn \varphi(\ol{x}) \leftrightarrow
    \exists \ol{y} . \psi(\ol{x}, \ol{y})
  .\]
  It is a $\Pi_1$-formula if there is a
  $\Delta_0$-formula $\psi(\ol{x}, \ol{y})$ such
  that
  \[
    \PA \syn \varphi(\ol{x}) \iff \forall \ol{y} .
    \psi(\ol{x}, \ol{y})
  .\]
\end{fcdefn}

In Example Sheet 4, you will prove that the
characteristic function of a $\delz$-definable set
is \gls{pr}. We will show that the
$\sigone$-definable sets are precisely the
\gls{renum} ones.

Recall that defining $\langle x, y \rangle =
\frac{(x + y)(x + y + 1)}{2} + y$ yields a
\gls{tr} bijection $\Nbb^2 \to \Nbb$.

Applying this a bunch of times, we get \gls{tr}
bijections $\Nbb^k \to \Nbb$ by $\langle v, \ol{w}
\rangle = \langle v, \langle \ol{w} \rangle \rangle$.

This is not good, as we have a different function
for each $k$. We'd like a ``pairing function''
that lets us see a number as a code for a sequence
of any length.

This can be done within any model of \gls{PA} by
using a single function $\beta(x, y)$ (known as
G\"odel's $\beta$-function) which is definable in
\gls{PA}.

We want an arithmetic procedure that can associate
a code to sequences of any length, and such that
the entries of the sequence can be recovered from
the code.

We will do this by a clever application of the
Chinese Remainder Theorem.
