Semantics encoding in the context of "Gödel numbering"

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⭐ Core Definition: Semantics encoding

A semantics encoding is a translation between formal languages. For programmers, the most familiar form of encoding is the compilation of a programming language into machine code or byte-code. Conversion between document formats are also forms of encoding. Compilation of TeX or LaTeX documents to PostScript are also commonly encountered encoding processes. Some high-level preprocessors, such as OCaml's Camlp4, also involve encoding of a programming language into another.

Formally, an encoding of a language A into language B is a mapping of all terms of A into B. If there is a satisfactory encoding of A into B, B is considered at least as powerful (or at least as expressive) as A.

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👉 Semantics encoding in the context of Gödel numbering

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. Kurt Gödel developed the concept for the proof of his incompleteness theorems.

A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic.

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Semantics encoding in the context of Self-reference

Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.

In natural or formal languages, self-reference occurs when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding.

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