Talk:Diagonal lemma/Proof with diagonal formula/Object language

Signature

The signature $\Sigma$ of the object language:

$\Sigma :\left(F+R\right)\to \mathbb {N}$ (only in this line, $+$ is meant for direct product!)

$F=\left\{0,s,+,\cdot \right\}$

$R=\emptyset$

$\Sigma \left(0_{\mathrm {fuctorname} }\right)=0$

$\Sigma \left(s_{\mathrm {fuctorname} }\right)=1$

$\Sigma \left(+_{\mathrm {fuctorname} }\right)=2$

$\Sigma \left(\cdot _{\mathrm {fuctorname} }\right)=2$

Axioms

For $\Gamma$, let us use Peano axioms.

Standard modell

Universe: $\mathbb {N}$, the set of natural numbers.

Interpretation:

$I\left(0_{\mathrm {fuctorname} }\right)=0_{\mathbb {N} }$

$I\left(s_{\mathrm {fuctorname} }\right)=s_{\mathbb {N} }$

$I\left(+_{\mathrm {fuctorname} }\right)=+_{\mathbb {N} }$

$I\left(\cdot _{\mathrm {fuctorname} }\right)=\cdot _{\mathbb {N} }$

where the right-hand sides are meant for zero, inrementation, addition and multiplication among the natural numbers.