Elementary function arithmetic

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, +, ×, ${\displaystyle x^{y}}$, together with induction for formulas with bounded quantifiers.

EFA is a very weak logical system, whose proof theoretic ordinal is ${\displaystyle \omega ^{3}}$, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.