Elementary function arithmetic

Definition

• two constants 0, 1,
• three binary operations +, ×, exp, with exp usually written as x^y,
• a binary relation symbol <.

• The axioms of Robinson arithmetic for 0, 1, +, ×, <
• The axioms for exponentiation: x⁰ = 1, x^y+1 = x^y × x.
• Induction for formulas all of whose quantifiers are bounded.

  Friedman's grand conjecture

Related systems

• One can omit the binary function symbol exp from the language, by taking Robinson arithmetic together with induction for all formulas with bounded quantifiers and an axiom stating roughly that exponentiation is a function defined everywhere. This is similar to EFA and has the same proof theoretic strength, but is more cumbersome to work with.
• There are weak fragments of second-order arithmetic called RCA and WKL that have the same consistency strength as EFA and are conservative over it for Π sentences, which are sometimes studied in reverse mathematics.
• Elementary recursive arithmetic is a subsystem of primitive recursive arithmetic in which recursion is restricted to bounded sums and products. This also has the same Π sentences as EFA, in the sense that whenever EFA proves ∀x∃y P, with P quantifier-free, ERA proves the open formula P, with T a term definable in ERA. Like PRA, ERA can be defined in an entirely logic-free manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under composition and projection of a finite number of basis functions, and thus only a finite number of defining equations are needed.