I am a little confused about why PMI cannot help us in proving that the intersection of an countable infinity of closed sets is closed. My confusion comes from the proof of this fact given in section 2.4 of the notes (bottom of page 29). The end of the proof asserts something of this form:
1. Something about "A sub j" is true for all j in the index set.
2. Then that something is true for the infinite intersection of all "A sub i."
It seems like PMI can prove statement 1 above as long as we are dealing with a countable infinity of closed A's. And If PMI can prove statement 1 (in the proof we were given, statement 1 was true because j was arbitrary), then we ought to be able to deduce statement 2. Here is why I think PMI can prove statement 1 for a denumerable case:
Let A1 denote "A sub 1," Ai denote "A sub i," and so on.
Let "A1 Int A2" denote "the intersection of A1 and A2," and so on.
Let the set of all Ai be infinite and be able to be indexed by Z+ (ie, we are dealing with a denumerable case). Also let all Ai be closed.
A1 Int A2 is closed, by Axiom C2.
A1 Int A2 Int A3 is closed, since "A1 Int A2" is closed (see previous step), and A3 is closed by hypothesis.
A1 Int A2 Int A3 Int A4 is closed... Unfortunately, it's too cumbersome to do a rigorous induction proof using only keyboard symbols, but clearly induction works in this case, so we CAN conclude that;
A1 Int A2 Int A3 ... Int An is closed for all n in Z+ (remember Z+ is the index set).
Here we seem to have proved something of the form of statement 1 at the top of this posting. Can't we conclude statement 2, and therefore be proving something about the intersection of an infinite number of sets?
In brief, the proof in section 2.4 seems to say: if such-and-such is true for all Ai, then it is true for the intersection of all Ai. But induction seems the perfect vehicle for proving that such-and-such is true of all Ai, as long as i is in Z+.
Any comments? Sorry this was so long!