Do Gödel numbers have a practical use? Is there any example of Gödel numbers being actually used in practice? If so for what purpose?
 A: The basic idea of Gödel numbers is to establish a mapping between logical statements and natural numbers. This allows to apply reasoning on natural numbers, and those results can be translated back into reasoning on logical statements.
Related mappings are: 


*

*the mapping between Turing machines, one embodiment of computable functions, and natural numbers. A main result is the Halting problem.
It influenced the recognition by engineers that compilers and software controlled systems can not be trusted fully, which plays a role in environments like atomic power plants.

*the mapping between logical statements and finite automata, which allows for model checking


So my impression is that the idea has had no direct practical application, like general relativity and GPS, but still was very influential to computer science.
A: The key idea of Gödel numbering is that logical syntax can be treated as data. Applications of this concept pervade modern computing.
A: Yes. The Information Age is based on the Gödel number.
A program in the programming language Python, for instance, is stored internally in the computer as a base-128 integer, via the ASCII which encodes characters as integers. If you wish to execute the program, you provide it to the computer as a Gödel number, expressed in base 128 (by typing in certain keys on your keyboard), and you tell the computer to execute the program with that Gödel number. You're not coding Turing machines directly, but you're coding a Turing-equivalent form of machine.
