nLab elementary function arithmetic

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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

semantics

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

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,

Last revised on September 6, 2012 at 20:35:48. See the history of this page for a list of all contributions to it.