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!)
 * $$F = \left\{0, s, +, \cdot\right\}$$
 * $$R = \emptyset$$
 * $$\Sigma\left(0_{\mathrm{fuctor name}}\right) = 0$$
 * $$\Sigma\left(s_{\mathrm{fuctor name}}\right) = 1$$
 * $$\Sigma\left(+_{\mathrm{fuctor name}}\right) = 2$$
 * $$\Sigma\left(\cdot_{\mathrm{fuctor name}}\right) = 2$$

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

Standard modell
Universe: $$\mathbb N$$, the set of natural numbers.

Interpretation:
 * $$I\left(0_{\mathrm{fuctor name}}\right) = 0_{\mathbb N}$$
 * $$I\left(s_{\mathrm{fuctor name}}\right) = s_{\mathbb N}$$
 * $$I\left(+_{\mathrm{fuctor name}}\right) = +_{\mathbb N}$$
 * $$I\left(\cdot_{\mathrm{fuctor name}}\right) = \cdot_{\mathbb N}$$

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