User blog comment:Wythagoras/All my stuff/@comment-10429372-20130715145943/@comment-7484840-20130716184105

@ikosaract1: yes, if the I function that is being extended is the one that diagonalises throughout the whole of time, then yes, it can be extended.

@FB100Z: but a function referring to a version of itself that cannot refer to itself is a single self referral and contains no paradoxes or infinities. A double self referral, where f(n) (a definability function) refers to f(m) which refers to f(o), where f(o) cannot refer to the f function and o>m>n, also contains no problems. Neither does a triple self-referral, a quadruple self-referral or so on. The problem is where to stop. Yes, it avoids infinities, but this just leaves it with the largest number that is not infinity as its output, which behaves just like infinity, except it isn't actually infinity.