Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?
I know that the Cantor–Bernstein–Schroeder theorem
implies the existence
of a 1-1 mapping between the reals and the irrationals,
but the proofs of this theorem
I wondered if a simple
(not involving an infinite set of mappings)
(so the mapping is straightforwardly specified)
I have considered
mapping the rationals
to the rationals plus a fixed irrational,
but then I could not figure out
how to prevent an infinite
(possible uncountably infinite)
Map numbers of the form q+k√2 for some q∈Q and k∈N to q+(k+1)√2 and fix all other numbers.