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Example (Continued)

From f, we remove all "doubles" constructing a sequence f': N -> Q that contains each positive rational number in exactly one position.

Finally we can define an enumeration of all rationals

g: N ->bijective Q

g(x) :=

if x = 0 then 0

else if x is even

then -f'(x/2)

else f'((x-1)/2)

g = [0, 1, -1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, ...].