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A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.
These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.
For example, John von Neumann constructs the number 0 as the empty set {}, and the successor of n, S(n), as the set n ∪ {n}. The axiom of infinity then guarantees the existence of a set that contains 0 and is closed with respect to S. The smallest such set is denoted by N, and its members are called natural numbers. [2]
In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 (the successor function) = n ∪ {n} for each n. In this way n = {0, 1, …, n − 1} for each natural number n. This definition has the property that n is a set with n elements.
Computable number: A real number whose digits can be computed by some algorithm. Period: A number which can be computed as the integral of some algebraic function over an algebraic domain. Definable number: A real number that can be defined uniquely using a first-order formula with one free variable in the language of set theory.
A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number.
Typographical Number Theory (TNT) is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach. It is an implementation of Peano arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems .
The unary numeral system is the simplest numeral system to represent natural numbers: [1] to represent a number N, a symbol representing 1 is repeated N times. [2]In the unary system, the number 0 (zero) is represented by the empty string, that is, the absence of a symbol.