What is the necessary condition for the process of "proceeding to the limit" in Whitehead and Russell's PM?

Question

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:

1. An nth-order function is one that takes (n-1)th and lower order functions as arguments.

2. A first order function is one that takes only "individuals" as arguments.

3. A second order function is one that take first order function and "individuals" as arguments.

4. 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.

Answer 1

Since the usual set-theoretic suspects don't seem to be responding I would like to comment briefly that you seem to be confusing two notions of limit. The limit you have in calculus can by all means be taken, and is taken many times every second if you try to estimate the number of calculus students around the globe. The "conditions" for taking such limits can be found in every calculus textbook.

Meanwhile, PM is talking specifically about a situation where one cannot take a "limit" where of course limit is understood in a more nebulous way. Namely, the types exist only for finite orders, and restricting them to finite orders may be precisely what is needed so as to escape the paradoxes of naive set theory that Russell is famous for spotting.