Definition 2.1.23 (Godel numbering). Let $L$ be a first-order language. A Gödel numbering is an injection $L\hookrightarrow \mathbb{N}$ that is:

• (1)
  Computable (assuming some notion of computability for strings of symbols over a finite alphabet);
• (2)
  Its image is a recursive subset of $\mathbb{N}$;
• (3)
  Its inverse (where defined) is also computable.