Gödel number

mathematics

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June 29, 2026

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The term "Gödel number" refers to a unique natural number assigned to each expression, formula, and proof within a formal system, as developed by mathematician Kurt Gödel. This numbering system was a crucial part of Gödel's groundbreaking incompleteness theorems, published in 1931.123

Gödel's first incompleteness theorem established that in any consistent formal system of mathematics that includes arithmetic, there will always be true statements that cannot be proven within that system. His second incompleteness theorem further demonstrated that such a system cannot prove its own consistency. These theorems fundamentally changed the understanding of mathematics, revealing inherent limitations in formal axiomatic systems and impacting the philosophy of mathematics.234

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set theory
Limitations of axiomatic set theory
…with natural numbers (now called Gödel numbers) and by talking about these numbers, Gödel was able to make the metamathematics of S become part of the arithmetic of S and hence expressible in S. The theorem in question asserts that the formula of S that expresses (via a coding) “S…
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…with natural numbers (now called Gödel numbers) and by talking about these numbers, Gödel was able to make the metamathematics of S become part of the arithmetic of S and hence expressible in S. The theorem in question asserts that the formula of S that expresses (via a coding) “S…
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automata theory
The generalized automaton and Turing’s machine
…a corresponding integer, called the Gödel number, that identifies the Turing machine capable of computing the given function. The Gödel number and the argument value of the function to be computed can be given as input data on the tape of the universal machine. From the Gödel number, the list…
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…a corresponding integer, called the Gödel number, that identifies the Turing machine capable of computing the given function. The Gödel number and the argument value of the function to be computed can be given as input data on the tape of the universal machine. From the Gödel number, the list…
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metalogic
Truth definition of the given language
…numbers that are just the Gödel numbers of the true sentences of N.
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…numbers that are just the Gödel numbers of the true sentences of N.
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