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Table 1 Gödel number representation. Original Gödel numbers cannot represent the Directed Acyclic Graph (DAG) structure of the Gene Ontology.

From: An ontology-based search engine for protein-protein interactions

Term Natural number Relation Gödel Number
is a 1   
part of 2 R1: Term2 is a Term1 243153 = 6, 000
Term1 3 R2: Term3 is part of Term2 253254 = 180, 000
Term2 4 R3: Term4 is a Term2 263154 = 120, 000
Term3 5   
Term4 6   
  1. The following example shows why the original Gödel numbers fail to represent the GO structure. Suppose that we represent terms by unique natural numbers and the relations between them by Gödel numbers. In this example, Term4 is a kind of Term2 by relation R3, and Term4 is a kind of Term1 by relation R1. But these relations cannot be inferred from the representation because the original Gödel numbers are not sufficient to represent the DAG structure of GO.