# The Undefined Value of Arithmetic

We want substitution of an expression for a free variable always be allowed

• sqrt(x) must not imply that we are only talking about non-negative reals
• otherwise the root 3 would be lost in the following:
11 sqrt(|x|+1) = 25 - x ⇔
(x < 0 ^^ 11 sqrt(1-x) = 25 - x)vv(x >= 0 ^^ 11 sqrt(x+1) = 25 - x)

Each undefined expression must yield a “value”

• we will see that one value suffices
• we denote it with _|_

Let f and g be expressions without free variables

• if f contains an undefined subexpression but f yields a for some a in RR,
then f = x has a wrong root x = a
arithmetic must be regular รก la Kleene:
undefined argument guarantees undefined result
• if f = g yields T although f or g is undefined,
then each x in RR is its wrong root
frac(1)(0) = frac(1)(0) does not yield T, so ”=“ is not reflexive!

We also want to solve f(x) > 0 and so on

if f or g is undefined, then none of f = g, f < g, f <= g, … yields T
• so we cannot distinguish between different undefineds, so a single _|_ suffices
(we could build that ability to the logic but we do not want to)