Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician Skolem...

In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all...

any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite...

is given by a primitive recursive relation (Smith 2007, p. 141). As such, the Gödel sentence can be written in the language of arithmetic with a simple...

mathematics (Simpson 2009). Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted...

mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic. The introduction of infinite mathematical objects occurred...

elementary recursive function, also called an elementary function, or a Kalmár elementary function, is a restricted form of a primitive recursive function...

interpretation of intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed...

IΣ1 of Peano arithmetic in which induction is restricted to Σ01 formulas. In turn, IΣ1 is conservative over primitive recursive arithmetic (PRA) for Π...

in medicine Positive relative accommodation Primitive recursive arithmetic, a formal system of arithmetic Probabilistic risk assessment, an engineering...

example, in primitive recursive arithmetic any computable function that is provably total is actually primitive recursive, while Peano arithmetic proves that...

Boolean Algebra, Pergamon Press 1963, Dover 2007 Recursive number theory - a development of recursive arithmetic in a logic-free equation calculus, North Holland...

(Peano arithmetic in this case) it can be proven that the theories ZFC+A and ZFC+B are equiconsistent. Usually, primitive recursive arithmetic can be...

primitive recursive arithmetic P R A {\displaystyle {\mathsf {PRA}}} . The theory may be extended with function symbols for any primitive recursive function...

Dialectica interpretation of intuitionistic arithmetic developed by Kurt Gödel. In recursion theory, the primitive recursive functionals are an example of higher-type...

carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle. Tait (2005) gives a game-theoretic...

the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. Moreover, Gödel's second incompleteness...

_{0}^{0}=\Pi _{0}^{0}=\Delta _{0}^{0}} , since using primitive recursive functions in first-order Peano arithmetic requires, in general, an unbounded existential...

used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular...

multiplication and equality. Primitive recursive arithmetic, a quantifier-free formalization of the natural numbers. True arithmetic, the statements true about...

expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complicated arithmetic expressions such...

induction present in arithmetics stronger than Q turns this axiom into a theorem. x + 0 = x x + Sy = S(x + y) (4) and (5) are the recursive definition of addition...

discovered an alternative form of primitive recursive arithmetic, known as elementary recursive arithmetic, based on primitive functions that differ from the...

references can occur. A process that exhibits recursion is recursive. Video feedback displays recursive images, as does an infinity mirror. In mathematics and...

founders of finitism in mathematics. Skolem (1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions...

4 Weaker systems than recursive comprehension can be defined. The weak system RCA* 0 consists of elementary function arithmetic EFA (the basic axioms...

natural number with a given property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. Suppose...

for the benefits of finitistic mathematical systems, such as primitive recursive arithmetic, which do not include actual infinity. A conference in honor...

successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the...

fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0...