Code

A code is a mathematical object, often a set of natural numbers, used to represent another object such as a countable structure or a set.

Gödel numbering: A systematic assignment of natural numbers to symbols, formulas, or proofs. It exemplifies how syntactic objects can be represented by numbers.
Natural number: A number used for counting, with conventions differing on whether zero is included. Natural numbers are the usual target objects for codes.
Gödel's incompleteness theorems: Results showing that sufficiently expressive consistent formal systems contain undecidable statements and cannot prove their own consistency. Gödel coding makes formal syntax available to arithmetic.
Gödel numbering of programs: An assignment of natural numbers to programs that supports effective manipulation of their descriptions. It is a specialized syntax code, unlike codes for arbitrary structures.
Characteristic function: A function that maps members of a set to 1 and nonmembers to 0. It converts a set into a numerical representation.
Countable set: A set that is finite or can be put in bijection with the natural numbers. Countability makes numerical representation possible.
Descriptive set theory: The study of definable sets in topological spaces, especially Polish spaces. Codes let this field study countable structures through sets of reals.
One-hot encoding: A representation that assigns each category a distinct vector with exactly one nonzero coordinate. It illustrates a finite-vector encoding rather than coding by natural-number sets.
Pairing function: A bijection that encodes pairs of natural numbers as single natural numbers. It lets codes represent structured data such as ordered pairs.
Bijection: A function that is both one-to-one and onto. Bijections formalize when a coding loses no information.

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