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The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers.In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition are defined. [1]
The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbers less than a given natural number is computable. The subset of prime numbers is computable. A recursive language is a computable subset of a formal language.
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. [1] [2] Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion ...
In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions.
This characterization states that a function is primitive recursive if and only if there is a natural number m such that the function can be computed by a Turing machine that always halts within A(m,n) or fewer steps, where n is the sum of the arguments of the primitive recursive function. [5]
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: There is an algorithm such that the set of input numbers for which the algorithm halts is exactly S. Or, equivalently,
This is an accepted version of this page This is the latest accepted revision, reviewed on 9 February 2025. General-purpose programming language "C programming language" redirects here. For the book, see The C Programming Language. Not to be confused with C++ or C#. C Logotype used on the cover of the first edition of The C Programming Language Paradigm Multi-paradigm: imperative (procedural ...
In Set, the category of sets, the standard natural numbers are an NNO. [6] A terminal object in Set is a singleton, and a function out of a singleton picks out a single element of a set. The natural numbers 𝐍 are an NNO where z is a function from a singleton to 𝐍 whose image is zero, and s is the successor function.