Gödel number

Author: Leandro Alegsa Creation date: November 13, 2022

In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). The concept was first used by Kurt Gödel for the proof of his incompleteness theorem.

A Gödel numbering can be interpreted as an encoding where a number is assigned to each symbol of a mathematical notation, and a stream of natural numbers can then represent some form or function. A numbering of the set of computable functions can then be represented by a stream of Gödel numbers (also called effective numbers). Rogers' equivalence theorem states criteria for which those numberings of the set of computable functions are Gödel numberings.

Questions and Answers

Q: What is a Gödel numbering?

A: A Gödel numbering is a function that assigns a unique natural number to each symbol and formula of a formal language, called a Gödel number (GN).

Q: Who first used the concept of Gödel numbering?

A: Kurt Gödel first used the concept of Gödel numbering for the proof of his incompleteness theorem.

Q: How can we interpret Gödel numbering?

A: We can interpret Gödel numbering as an encoding where each symbol of a mathematical notation is assigned a number, and a stream of natural numbers can represent some form or function.

Q: What do we call the natural numbers assigned by a Gödel numbering?

A: The natural numbers assigned by a Gödel numbering are called Gödel numbers or effective numbers.

Q: What does Rogers' equivalence theorem state?

A: Rogers' equivalence theorem states criteria for which those numberings of the set of computable functions are Gödel numberings.

Q: What is represented by a stream of Gödel numbers?

A: A numbering of the set of computable functions can be represented by a stream of Gödel numbers.

Q: Why is Gödel numbering important in formal number theory?

A: Gödel numbering is important in formal number theory as it provides a way to represent mathematical formulas and functions as natural numbers, which allows for the proof of important theorems like the incompleteness theorem.