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In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, xy, together with induction for formulas with bounded quantifiers.

EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.


EFA is a system in first order logic (with equality). Its language contains:

Bounded quantifiers are those of the form ∀(x < y) and ∃(x < y) which are abbreviations for ∀ x (x < y) → ... and ∃x(x < y)∧... in the usual way.

The axioms of EFA are

Friedman's grand conjecture

Harvey Friedman'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 Friedman (1999) is:

"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."

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:

See also