User blog comment:Testitemqlstudop/On mapping positive integers to ordinals/@comment-43798125-20191027101113/@comment-35470197-20191028033231

> You cannot map it to O(log n), since the inverse function ord^-1 is impossible, with your fix or not.

You are mistaking here. Since we have no way to directly deal with ordinals, the "order" in the computation theory literary does not make sense.

Namely, you should replace the value of the given function (ord here) by the corresponding expression, and need to consider the way to compare two expressions. (For example, w^0+0 and w^{0+0}+0+0 are different expressions, but corresponds to the same ordinals.)

I agree that it is non-trivial to create a bijective computable map from a surjective computable map. Just modifying by adding 1 is valid only when he or she already coded the comparison algorithm for expressions, which is essentially equivalent to code a bijective map.