All the known bases of Combinatory Logic, such as {S,K}, or {K,W,B,C} have one or more combinators using 3 variables:
S = λxλyλz. x z(y z)
B = λxλyλz. x (y z)
C = λxλyλz. x z y
This begs the question whether we can make a basis with functions of only 2 variables. Surely if such a thing were possible, it would be a most interesting result and it would be well-known. But as far as I can tell, no such thing is known. Thus it seems nobody believes such a thing to be possible.
But has this been proven anywhere?
How do we know that {K,W,2,O,T,D} is not a basis, where
K = λxλy. x
W = λxλy. x y y
2 = λfλx. f(f x)
O = λxλy, y(x y)
T = λxλy. y x
D = λx. x x
?