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.