nLab
recursive subset

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Deduction and Induction

Foundations

Contents

Idea

A set SS of natural numbers is recursive if there is an algorithm which will decide in finitely many steps whether a given natural number belongs to SS.

Definition

A subset SS of the set of natural numbers \mathbb{N}, or more generally of k\mathbb{N}^k with kk finite, is recursive if there is a computable function (a total recursive function) f: k2={0,1}f: \mathbb{N}^k \to \mathbf{2} = \{0, 1\} \subseteq \mathbb{N} such that S=f 1(1)S = f^{-1}(1). Recursive subsets are a proper subclass of the class of recursively enumerable? sets, which are domains of partial recursive functions f: kf: \mathbb{N}^k \to \mathbb{N}, or equivalently images of total recursive functions.

Properties

Recursive sets form a Boolean subalgebra of the power set algebra P()P(\mathbb{N}) (whereas recursively enumerable subsets do not form a Boolean subalgebra). Even in constructive mathematics (where the power set algebra may be only a Heyting algebra), the recursive sets form a Boolean algebra (a Boolean subalgebra of the algebra of decidable subsets).

Examples and counterexamples

  • One can encode proofs in the formal theory PAPA (Peano arithmetic) as natural numbers, via a process of Gödel-numbering?. The set of codes of such formal proofs is a recursive set. In colloquial language: it is possible to program a computer to detect whether or not a string of symbols represents a valid proof in PAPA.

  • It follows that the set of codes of theorems (provable propositions) is recursively enumerable. However, it is not recursive. This is one way of saying that provability in PA cannot be decided by an algorithm, which is closely related to Gödel’s incompleteness theorems.

Last revised on August 13, 2013 at 17:11:41. See the history of this page for a list of all contributions to it.