nLab theorem





The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

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


There are good reasons why the theorems should all be easy and the definitions hard. (Michael Spivak, preface to “Calculus on Manifolds” )



In the traditional language of mathematics, a theorem is a statement which is of interest in its own right and which has been proven to be true, though the proof may not be immediately obvious. This contrasts with a lemma (which is usually of interest primarily because of its implications for other statements), a conjecture (which has not yet been proved), an axiom (which is obviously true or assumed to be true), a definition (which becomes true by virtue of its assigning meaning to a word or phrase), a proposition (which usually follows more easily from known facts than a theorem does), or a corollary (which follows immediately from facts recently proven).

The discipline of logic formalizes the notion of proof, but not the notions of “interest” or “immediacy”. Thus, to a logician, any proved statement is often called a theorem. (Mathematicians know this meaning too, but still usually reserve the term ‘theorem’ for important theorems in their published work.) The term ‘proposition’, to a logician, means any statement and does not imply the existence of a proof. The term ‘axiom’ is used in a way that somewhat matches its ordinary usage, but as a logician counts trivial proofs as proofs, an axiom is also a special case of a theorem. Logic rarely studies definitions explicitly, but in some theories they do play a role, similar to their informal usage. The other terms appear not to be used in logic.


In a given logic, in a given context, we have various propositions and various proofs of propositions. In that context, a theorem is a proposition with a proof.

Classically, a theorem is a proposition for which there exists a proof, but in some contexts (such as, perhaps, fully formalized constructive type theory), one may use “theorem” to mean a proposition together with a proof.

A theorem should be contrasted with a tautology: a proposition that is true in all models. If every theorem in a given logic is a tautology in a given class of models for that logic, then we say that the class of models is sound for that logic; if conversely every tautology is a theorem, then we say that the class of models is complete.


… we might list famous important theorems/lemmas/etc in the nnLab here …


  • A mathematician is a device for turning coffee into theorems. —Alfréd Rényi

  • Lemmas do the work in mathematics: Theorems, like management, just take the credit. —Paul Taylor

mathematical statements


  • Thomas Hales, Formal proof (pdf)

  • John Harrison, Formal proof – theory and practice (pdf)

Last revised on March 5, 2023 at 13:32:11. See the history of this page for a list of all contributions to it.