I think in untyped lambda calculus you work with expressions, so you take your f function (somehow represented as an expression in untyped lambda calculus), you put that in a fixed point combinator, and the resulting expression is a fixed point of your f function.

It most certainly won't be a representation of a number->number function, but a general element of untyped lambda calculus nevertheless.

It is somehow similar how you take square root of a negative number on the Complex plane. You represent your negative number as a complex number (r \mapsto r + 0i), then you can take a square root of it, but that won't correspond to a representation of a real number.

Using church numerals[1],

  right_incr(n)  =
    g => y => n(g)(g(y))

and using the following shorter fixed-point combinator[2],

  X(f) =
    ( a => a(a) )( x => f(x(x)) )

I get this junk:

  X(right_incr)
    = ( a => a(a) )( x => g => y => x(x)(g)(g(y)) )
    = ( x => g => y => x(x)(g)(g(y)) ) ( x => g => y => x(x)(g)(g(y)) )

  X(right_incr)(g)(y)
    = let f = ( x => g => y => x(x)(g)(g(y)) )
      in f(f)(g)(y)
    = let .. in f(f)(g)(g(y))
    = let .. in f(f)(g)(g(g(y)))
    = ...

Which, as you can see, only gets more complex when I try to transform it.

[1] https://en.m.wikipedia.org/wiki/Church_encoding [2] https://en.m.wikipedia.org/wiki/Fixed-point_combinator#Other...