PlanetMath: Elementary Functional Arithmetic

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Elementary Functional Arithmetic

(Definition)

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)
$\forall x,y(x<y'\leftrightarrow x<y\vee x=y)$
$\forall x(x^0=1)$
$\forall x(x^{y'}=x^y\cdot x)$

"Elementary Functional Arithmetic" is owned by Henry.

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