User blog comment:D57799/Some discoveries and guesses about uparrow notation/@comment-5529393-20150208214728

Just to add to LittlePeng9's point, we can't define tetration on the rationals satisfying the properties in 1. because the values vary greatly depending on the numerator and denominator of the rational number. For example, 2^^2 = 4, but (2^^(4/2)) = (2^^4)^^1/2 = 65536^^1/2 > 6. More generally a^^(2k/k) > a^^k, which goes to infinity as k goes to infinity, rather than equalling a^^2.