[FOM] Fractional Iteration and Rates of Growth

[ali enayat]

This is a reply to Dmytro Taranovsky's posting on May 1, 2006, dealing with 
natural analogues of primitive recursive functions defined on the set of 
positive reals.

The typical scenario in a foundations of analysis course is to extend the 
addition, multipication, and exponentiation functions from the set of 
natural numbers, first to the set of integers, then to the set of rational 
numbers, and finally, by relying on  appropriate limit arguments, to the set 
of reals (or at least positive reals, e.g., for a^x, where a is required to 
be positive).

As it turns out, one encounters highly nontrivial difficuluties in extending 
*even the super exponentiation function* to the set of positive real 
numbers. I suggest consulting the following URL for a taste of the state of 
the art:


http://en.wikipedia.org/wiki/Tetration

Best regards,

Ali Enayat