nLab
elementary function arithmetic

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
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, (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

Foundations

Elementary function arithmetic

Idea

Elementary function arithmetic, or EFA, is a first-order theory of natural numbers, one of the weakest fragments of arithmetic strong enough to do nontrivial mathematics. It is strictly weaker than Peano arithmetic?.

Regarding the strength of EFA, Harvey Friedman has put forth the following “grand conjecture”:

“Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, ×, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded.”

Here “bounded quantifier” refers to a quantifier of shape m<n\forall_{m \lt n}; more formally, if a variable nn does not occur in a formula ϕ\phi, then m<nϕ(m)\forall_{m \lt n} \phi(m) means m(m<n)ϕ(m)\forall_m (m \lt n) \Rightarrow \phi(m).

Definition

The language of EFA is one-sorted with

  • Two constants 0,10, 1

  • Three binary function symbols +,,exp+, \cdot, \exp (where exp(x,y)\exp(x, y) is usually written x yx^y).

The axioms of EFA are

  • Those of Robinson arithmetic (where ss is translated as 1+()1 + (-), i.e., as +(1,)+(1, -));

  • Exponentiation axioms, viz. x 0=1x^0 = 1 and x yx=x y+1x^y \cdot x = x^{y+1};

  • The induction axiom for formulas all of whose quantifiers are bounded.

References

“Grand conjectures” by Harvey Friedman may be found here:

A MathOverflow discussion on Friedman’s grand conjecture about EFA may be found here,

Revised on September 6, 2012 20:35:48 by Toby Bartels (98.23.143.147)