Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic.

Quines are possible in any programming language that has the ability to output any computable string, as a direct consequence of Kleene's recursion theorem.

A corollary to Kleene's recursion theorem states that for every Gödel numbering of the computable functions and every computable function, there is an index such that returns.

By the corollary to the recursion theorem, there is an index such that returns.

A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star ( Kleene closure ), Kleene's recursion theorem and the Kleene fixpoint theorem.

The second recursion theorem is closely related to definitions of computable functions using recursion.

Because it is better known than the first recursion theorem, it is sometimes called just the recursion theorem.

: The second recursion theorem.

The theorem does not guarantee that e is an index for the smallest fixed point of the recursion equations ; this is the role of the first recursion theorem described below.

One informal interpretation of the second recursion theorem is that any partial computable function can guess an index for itself.

The recursion theorem shows that no computable function is fixed point free, but there are many non-computable fixed-point free functions.

The first recursion theorem is related to fixed points determined by enumeration operators, which are a computable analogue of inductive definitions.

: First recursion theorem.

Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations.

DC is the fragment of AC required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step.

These images, such as of his canonical Mandelbrot set pictured in Figure 1 captured the popular imagination ; many of them were based on recursion, leading to the popular meaning of the term " fractal ".

Some special purpose languages such as Coq allow only well-founded recursion and are strongly normalizing ( nonterminating computations can be expressed only with infinite streams of values called codata ).

Information Processing Language was the first AI language, from 1955 or 1956, and already included many of the concepts, such as list-processing and recursion, which came to be used in Lisp.

In languages such as Prolog, mutual recursion is almost unavoidable.

A conservative position, taken by Lyle Campbell, is that it would have shared the " design features " of known human languages, such as grammar, defined as " fixed or preferred sequences of linguistic elements ", and recursion, defined as " clauses embedded in other clauses ", but that beyond this nothing can be known of it ( Campbell and Poser 2008: 391 ).

As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is that given a set g < sub > 1 </ sub >, and class functions G < sub > 2 </ sub >, G < sub > 3 </ sub >, there exists a unique function F: Ord → V such that

More generally, one can define objects by transfinite recursion on any well-founded relation R. ( R need not even be a set ; it can be a proper class, provided it is a set-like relation ; that is, for any x, the collection of all y such that y R x must be a set.

Most mathematicians, including recursion theorists, use the term " domain of f " for the set X < nowiki >'</ nowiki > of all values x such that f ( x ) is defined.

Forcing was considerably reworked and simplified in the 1960s, and has proven to be an extremely powerful technique both within set theory and in areas of mathematical logic such as recursion theory.

In this context, a fixed point of F is an index e such that ; the function computed by such an e will be a solution to the original recursion equations.

The first recursion theorem ( in particular, part 1 ) states that there is a set F such that Φ ( F )

Chomsky posited humans possess a special, innate ability for language and that complex syntactic features, such as recursion, are " hard-wired " in the brain.

More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics.

The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.

Compilers may also save more information in the recursion stack than is strictly necessary, such as return address, unchanging parameters, and the internal variables of the procedure.

The 91 function was chosen for having a complex recursion pattern ( contrasted with simple patterns, such as defining by means of ).

This choice is motivated by the fact that in generalized recursion theories, such as α-recursion theory, the definition corresponding to domains has been found to be more natural.

Systems with explicit recursion combinators, such as Plotkin's PCF, are not normalizing, but they are not intended to be interpreted as a logic.

In particular, it is easy to define both control structures such as recursion, loops and sequential composition and datatypes such as first-order functions, truth values, lists and integers.

These objects can control sprites and other media remotely, without being attached to any one sprite, may be used to control data or other non-displayed items, and are useful for recursion routines such as pathfinding.

The two recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions.

The restriction to recursion equations that can be recast as recursive operators ensures that the recursion equations actually define a least fixed point.

Thus there is no fixed point g satisfying these recursion equations.

One difference between the first and second recursion theorems is that the fixed points obtained by the first recursion theorem are guaranteed to be least fixed points, while those obtained from the second recursion theorem may not be least fixed points.

* Denotational semantics, where another least fixed point theorem is used for the same purpose as the first recursion theorem.

Stack overflow may be difficult to avoid when using recursive procedures, since many compilers assume that the recursion stack is a contiguous area of memory, and some allocate a fixed amount of space for it.

Alternatively, one can employ large base cases that still use a divide-and-conquer algorithm, but implement the algorithm for predetermined set of fixed sizes where the algorithm can be completely unrolled into code that has no recursion, loops, or conditionals ( related to the technique of partial evaluation ).