I read this from Introduction of the 1st edition of Principia Mathematica by Whitehead and Russell:
Since the orders of functions are only defined step by step, there can be no process of "proceeding to the limit," and functions of infinite order cannot occur.
Here is a summary of orders of functions:
An nth-order function is one that takes (n-1)th and lower order functions as arguments.
A first order function is one that takes only "individuals" as arguments.
A second order function is one that take first order function and "individuals" as arguments.
By "individuals" we mean constituent objects that are neither a function or a proposition and will not disappear like a class or a set after analysis.
At this point, I think an nth-order function is perfectly eligible for mathematical induction. I can't see the connection between something defined step by step and its inability to "proceed to the limit." It seems that the authors assume the readers are familiar with some well-known rules that specify the necessary condition for the process of "proceeding to the limit." I wonder what these rules are.