Gödelization

2020 Mathematics Subject Classification: Primary: 68P05 [MSN][ZBL]

A Gödelization or Gödel numbering of a $\Sigma$-algebra $A$ is a pair $(C,\beta)$ with $C$ as a recursive $\Sigma$-number-algebra and $\beta\colon A \longrightarrow C$ as a $\Sigma$-algebra-morphism from $A$ onto $C$. As usual, $\Sigma$ designates a signature. A Gödelization could be understood as encoding of the elements of $A$ by natural numbers [W90].

Gödelizations with bijective mapping $\beta$ are of special interest. They are the counterpart of a certain bijective coordinatisation mapping $\nu\colon C \longrightarrow A$. More precisely, $\beta=\nu^{-1}$ is the inverse of $\nu$ in this case. For a sensible signature $\Sigma$ and a term-generated $\Sigma$-algebra $A$, such a coordinatization with bijective mapping $\nu$ always exists. Taking together, every Gödelization induces canonically a corresponding coordinatization and vice versa.

The correspondence between Gödelization and coordinatization can be extended to congruences defined on $A$ and $C$. A congruence $\sim^\beta \subseteq A \times A$ induces a congruence $\sim^\nu \subseteq C \times C$ and vice versa via $\beta$ resp. $\nu$ [W90].

As one can easily see, a Gödelization with bijective mapping $\beta$ does not always exist. As a simple counterexample take a $\Sigma$-algebra $A$ with an uncountable number of elements.

References

[W90]

M. Wirsing: "Algebraic Specification", in J. van Leeuwen: "Handbook of Theoretical Computer Science", Elsevier 1990