Re: Realizability and Partial Combinatory Algebras

[<jvoosten@math.uu.nl>]

Dear John,

> Note that we can find elements if,true,false \in A satisfying
> 	if true x y = x, 	if false x y = y,
> and furthermore we can arrange that true = k.
> (I'll use "if then else" syntax below).
> We want to construct S such that (in A),
>      Skxkykz ~ (xkz)k(ykz), and Skxky is always defined.
> Take S to be
> 	\lambda txuyvz. if v then (xkz)k(ykz) else false
> using the usual Curry translation of lambda terms. To see that
> Skxky is always defined, note that, provably, Skxky(false)z = false.

There is a point I don't understand. If I work out Skxky, then I find
a term which contains subterms of the form xk, yk. So how can this be
defined if x or y represent nowhere defined functions?

It looks to me, that you are using a form of Combinatorial Completeness
that is not valid with the weaker S-axiom. If I am correct, one has
the following form of Combinatorial Completeness:

For every term t and variable z, there is a term \lambda z.t with the
property:

If \lambda z.t is defined, then for each a, (\lambda z.t)a ~ t[a/z]

(of course, a really correct version has more than one variable)

Best, Jaap