Primitive recursive arithmetic information

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...

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...

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

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

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

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

that exhibits recursion is recursive. In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined...

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

function can be chosen to be injective. The set S is the range of a primitive recursive function or empty. Even if S is infinite, repetition of values may...

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...

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

defined by a single primitive recursive function. A set X of natural numbers is defined by a formula φ in the language of Peano arithmetic (the first-order...

axioms, and recursively defined arithmetical operations. Fratres Bocca. pp. 83–97. Van Oosten, Jaap (June 1999). "Introduction to Peano Arithmetic (Gödel Incompleteness...

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...

theorem, which states that no recursive, consistent set of non-logical axioms Σ {\displaystyle \Sigma } of the Theory of Arithmetic is complete, in the sense...

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

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

first-order arithmetic which adopts that schema is denoted I Σ 1 {\displaystyle {\mathsf {I\Sigma }}_{1}} and proves the primitive recursive functions total...

means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from...