User blog comment:Kyodaisuu/Mashimo function/@comment-24509095-20140705233241/@comment-5150073-20140707153902

The "gap" has subjective meaning, and the best I can propose is take two numbers (or ordinals), put them in functions which define them and define the gap as f(n) and f(m) as |n-m|.

I call it subjective because there can be different functions defining two numbers: say, we want to compute the gap between $$10^{10^{100}}$$ and $$10^{10^{101}}$$. We can take the first as f(100) and the last as f(101) and say that the gap is "1", but as well we can take the first as f(10^100) and the last as f(10^101) and the gap would be 9*10^100. Probably the key is finding the strongest reasonable function somehow.