We've had surprising success embedding normal arithmetic in the Lambda
Calculus, modeling the natural numbers, addition, multiplication, and
so on. But one thing that some versions of arithmetic supply is a
-notion of infinity, which we'll write as `inf`. This object usually
+notion of infinity, which we'll write as `inf`. This object sometimes
satisfies the following constraints, for any finite natural number `n`:
n + inf == inf
n ^ inf == inf
leq n inf == true
- (Note, though, that with *some* notions of infinite numbers, like [[!wikipedia ordinal numbers]], operations like `+` and `*` are defined in such a way that `inf + n` is different from `n + inf`, and does exceed `inf`.)
+ (Note, though, that with *some* notions of infinite numbers, like [[!wikipedia ordinal numbers]], operations like `+` are defined in such a way that `inf + n` is different from `n + inf`, and does exceed `inf`; similarly for `*` and `^`. With other notions of infinite numbers, like the [[!wikipedia cardinal numbers]], even less familiar arithmetic operations are employed.)
9. Prove that `add ξ 1 <~~> ξ`, where `ξ` is the fixed
point you found in (1). What about `add ξ 2 <~~> ξ`?