Question on a transfinite construction in algebra I am studying out of Matsumura's Commutative Ring Theory, and in the first section on modules he proves (following Kaplansky) that every projective module over a local ring is free.
My questions have more to do with an application of transfinite induction than the actual algebra.
In the proof of the above result, Matsumura uses a lemma, and it is the proof of the lemma that I have a question on. The proof can be seen at Google Books: link. I am referencing Lemma 1 on page 10.
In the lemma, we define a family of submodules $\{F_\alpha\}$ using transfinite induction. I can follow the proof, i.e., I understand the (very) basic mechanics of transfinite induction and how each $F_\alpha$ is constructed and why the desired result follows.
Here are my questions:
(1) For every ordinal $\alpha$, we define a submodule $F_\alpha$. The ordinals do not form a set. Is it then true that I cannot speak of the set of all submodules $F_\alpha$? 
Matsumura writes about half way down the page that "if $F_\beta = F$ then the construction stops at $F_\beta$." Must this eventually happen? Must this construction terminate?
I am having a hard time wrapping my mind around the fact that we have modules (which are sets) and submodules (which are sets), and then all of the sudden we have a family of submodules which isn't (if I have all this right) a set. 
(2) Why can I not define a sequence of submodules over $\omega$? Why do I need the full strength of transfinite induction? Why do I need all ordinals?
As is obvious, this transfinite stuff is quite new to me; I have never used these techniques before. I would appreciate any insight.
 A: The idea is to generalize the idea of induction because some things require more than just a countably infinite sequence of steps in order to construct them.
For example, Borel sets are constructed by a transfinite induction and the construction is of length $\omega_1$, that is to say that we do not exhaust this construction until we exhausted all the countable ordinals.
Secondly, the ordinals are simply used to index the modules. $F_\alpha$ is merely a subset of $F$. We can talk on the set of subsets of $F_\alpha$ which are submodules. This is like saying $A_n=\text{some set of real numbers}$ and then talking about $A_\omega=\bigcup A_n$. Does that imply that $A_n$ is a set of natural numbers? Not at all.
Lastly, these sort of construction are usually of the form $F_\alpha\subsetneq F_{\alpha+1}\subsetneq\ldots$ and so if the construction does not stop we can define the function $\alpha\mapsto F_{\alpha+1}\setminus F_\alpha$ which is injective (do you see why?) and its range is a subset of $\mathcal P(F)$. This means that $F$ is a set whose power "set" is actually a proper class which is a contradiction to the power set axiom (which says that the power set of a set is indeed a set).
