This is a reply to Post QXQ countable sets which the system won't let me rreply to.

let g(m,n) = (2m-1)2^(n-1)

Is this a 1-1 mapping to N? If it is, it proves the rational numbers are countable since (m,n) can be interpreted as rational numbers.

g maps (m,n) to a unique even positive integer P. OK

Given P, is there a unique (m,n) st P=2N=(2m-1)2^(n-1), or,

N=(2m-1)2^(n-2), for any N? Obviously not because the left side can be even or odd and the right side is even for n>2.