What is the necessary condition for the process of "proceeding to the limit" in Whitehead and Russell's PM? 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.
 A: 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.
A: Short answer:
Mathematical induction can never proceed to infinity. The necessary condition for mathematical induction is that the property in question is possessed by an inductive number and is hereditary with respect to the relation $+_c1$.

Long answer:
This book is like building a tower from ground up without scaffolding,
  crane or any outside support. Everything that touches the ground must
  be explicitly spelt out and every brick must be supported by the ones
  from beneath not above. In other words, primitive ideas and primitive
  propositions are all the foundations this book has.* And every
  proposition is either a primitive one or must be proved by numbers
  less than itself. The formulae of induction are not reached until
  ✳91·17·171·373. The hierarchy of types is introduced in ✳12 which precedes the formulae of induction.
✳91·17 $\vdash:. P \in Potid‘R:\phi S. \supset_S. \phi(S|R):\phi(I {\restriction C‘R}): \supset \phi P $
Which states that if the property $\phi$ is hereditary with respect to $|R$, then if $\phi$ belongs to $I \restriction C‘R$ it belongs to any member of $Potid‘R$
Based on summary of Part III Section C, mathematical induction is a
  definition rather than a principle. In other words, some cardinals are
  inductive, some other cardinals are non-inductive. The defining
  property of inductive cardinals is this: An inductive cardinal is one
  which obeys mathematical induction starting from $0$, i.e. it is one
  which possesses every property possessed by $0$ and by the numbers
  obtained by adding $1$ to numbers possessing the property. In other
  words:
$\vdash :: \alpha \in NC induct .≡:. \xi \in \mu .\supset_\xi. \xi +_c
 1 \in \mu : 0 \in \mu \supset_\mu. \alpha \in \mu $
An inductive number by definition is one which can be reached from $0$
  by successive additions of $1$. Such a process can never reach
  infinity because ✳123 shows that the smallest infinite number $\aleph_0$ is not an inductive number.

*Due to the speculative nature of philosophy, the authors admit that "there must always be some elements of doubt, since it is hard to be
 sure that one never uses some principle unconsciously."  --see Part I.
 Section A, on first page of theory of deduction.
