|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
In informal mathematical speech and writing, a numeral refers to any notation or terms which denotes a natural number directly. Usually, in mathematics, this refers to base ten place-value notation, so that for instance , , , and are numerals.
In the more formal world of logic and type theory, a numeral generally means (e.g. Martin-Löf 73) a term whose type is a natural numbers type and which is of canonical form. The meaning of “canonical form” may vary with the formal theory, but with the usual presentation of the natural numbers as an inductive type generated by zero and the successor operation , the numerals are the terms of the form
Often, the numeral representing the natural number — which is to say, the term with applied times to — is denoted by . Thus, for instance, means . It is important to note that is not (usually) a term inside the formal system being considered; it is a “meta-notation” which stands for the term . (One might say that the underline converts informal numerals to formal ones.) In particular, any statement which quantifies over a natural number that occurs in a term can only be expressed in the metatheory?.
Not every term of natural number type is a numeral; consider for instance . However, good formal systems have the property of canonicity, which in this context means that every term of type computes to, or is provably equal to, a numeral. In our example, if is defined by recursion, there is a sequence of beta-reduction steps leading from to . (Canonicity is about terms in the empty context?; in a context with free variables of type , then of course there will be more terms of type , built out of these variables.)
If we add to such a formal system an axiom using an existential statement, then this is equivalent to adding to the language an additional term for a natural number that is not (and may not provably be equal to) any canonical numeral. For example, in (Peano arithmetic plus the axiom that Peano arithmetic is inconsistent?), we have the axiom
then (assuming that and so is in fact consistent) one can prove
in this system for every natural number , but one cannot prove
(which is actually trivially refutable).
A closed normal term with type symbol , which obviously must have the form , is called a numeral.