Elementary Functional Arithmetic, or EFA, is a weak theory of arithmetic created by removing induction from Peano Arithmetic. Because it lacks induction, axioms defining exponentiation must be added.

• $\forall x (x'\neq 0)$ ($0$ is the first number)
• $\forall x,y (x'=y'\rightarrow x=y)$ (the successor function is one-to-one)
• $\forall x (x+0=x)$ ($0$ is the additive identity)
• $\forall x,y(x+y'=(x+y)')$ (addition is the repeated application of the successor function)
• $\forall x(x\cdot 0=0)$
• $\forall x,y(x\cdot(y')=x\cdot y+x$ (multiplication is repeated addition)
• $\forall x(\neg (x<0))$ ($0$ is the smallest number)
• $\forall x,y(x<y'\leftrightarrow x<y\vee x=y)$
• $\forall x(x^0=1)$
• $\forall x(x^{y'}=x^y\cdot x)$