Elementary function arithmetic


In proof theory, a branch of mathematical logic, elementary function arithmetic, also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, , together with induction for formulas with bounded quantifiers.
EFA is a very weak logical system, whose proof-theoretic ordinal is, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.

Definition

EFA is a system in first-order logic. Its language contains:
  • two constants,,
  • three binary operations,,, with usually written as,
  • a binary relation symbol .
Bounded quantifiers are those of the form and which are abbreviations for and in the usual way.
The axioms of EFA are
  • The axioms of Robinson arithmetic for,,,,
  • The axioms for exponentiation:,.
  • Induction for formulas all of whose quantifiers are bounded.

    Friedman's grand conjecture

's grand conjecture implies that many mathematical theorems, such as Fermat's Last Theorem, can be proved in very weak systems such as EFA.
The original statement of the conjecture from is:
While it is easy to construct artificial arithmetical statements that are true but not provable in EFA, the point of Friedman's conjecture is that natural examples of such statements in mathematics seem to be rare. Some natural examples include consistency statements from logic, several statements related to Ramsey theory such as the Szemerédi regularity lemma, and the graph minor theorem.

Related systems

Several related computational complexity classes have similar properties to EFA:
  • 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 and that are conservative over EFA for sentences In particular, they are conservative for consistency statements. These fragments 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 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.