inhabited set


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 rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator 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

homotopy levels




In set theory, an inhabited set or occupied set is a set that contains an element.

More generally, in type theory an inhabited type is a type that has a term.

At least assuming classical logic, this is the same thing as a set that is not empty. Usually inhabited sets are simply called ‘non-empty’, but the positive word ‘inhabited’ reminds us that this is the simpler notion, which emptiness is defined as the negation of.

The terms ‘inhabited’ and ‘occupied’ come from constructive mathematics. In constructive mathematics (such as the internal logic of some topos or generally in type theory), a set/type that is not empty is not already necessarily inhabited. This is because double negation is nontrivial in intuitionistic logic. All the same, many constructive mathematicians use the old word ‘non-empty’ with the understanding that it really means inhabited.

There is a distinction between ‘inhabited’ and ‘occupied’ spaces in Abstract Stone Duality (which probably corresponds to something about locales, should explain that here).

An inhabited set is the special case of an inhabited object in the topos Set.

Revised on September 26, 2012 10:09:14 by Urs Schreiber (