User blog comment:Kyodaisuu/Mashimo function/@comment-1605058-20140706065714/@comment-25418284-20140706190939

I think if you used oracle Turing machines rather than ordinary ones, it will work.

Another solution is to use the FS definitions given at Talk:Rayo's number. Unfortunately, Rayo's function itself raises fundamental problems &mdash; its behavior depends on the mathematical philosophy you accept. (As a very strict formalist, I believe Rayo's function is ambiguously defined since no axiomatic theory was explicitly selected. Peng is a Platonist, and Rayo's function is consistent with his philosophy. You can see the kinds of issues we run into at this level of mathematics...)