nLab recursive subset



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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) 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


Deduction and Induction



The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



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.


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.


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 July 20, 2023 at 14:02:10. See the history of this page for a list of all contributions to it.