PROFESSOR: So we come to the part that a lot of students
have been asking about, but which
in fact is entirely optional.
So that if you care to skip this little piece of video,
you're welcome to.
It's not going to appear on any exam or anything.
But people have consistently asked
how they choose which method of proof
to use among ordinary induction or strong induction
or well-ordering.
And the answer is that it's hard to tell them apart,
because in an easy technical sense,
they're really all equivalent.
So let's look at them one by one.
First of all, it's clear that ordinary induction
is a special case of strong induction.
In the ordinary induction, you're
allowed to assume only p of n.
In strong, you can assign everything from p of 0
up to p of n to prove p of n plus 1.
But you don't have to use all the extra assumptions.
You could just use p of n so that any ordinary induction
can be seen as just a special case of a strong induction.
It would be a little misleading to call it strong induction,
but it is strong induction.
So why bother with it?
Well, the answers, basically, it's an expository difference.
It helps your reader to know that the proof for n plus 1
is only going to depend on n not on the k's that are less than n
as they would typically in a genuine strong induction proof.
Second, is some argument that an ordinary induction going from n
to n plus 1 is more intuitive than strong induction that
goes from anywhere less than or equal to n up to n plus 1.
I'm not sure that I subscribe to that,
but I've heard people make that claim.
All right.
There's another perspective, which
is interesting and maybe surprising, which is,
why not always use ordinary induction?
Oh, wait a minute.
How do you replace strong induction
with ordinary induction?
Well it's easy.
Suppose that you've proved for all m P of m
using strong induction with induction hypothesis P of m,
what have you done?
Well, it's the same base case whether you're
using ordinary or strong.
But in strong, you would do an inductive step
where you actually assumed not just p of n, but P of for all k
less than or equal to n.
And then using all those hypotheses about P of k,
you prove P of n plus 1 in the strong induction.
Well, how do you turn it into an ordinary induction?
Just let Q of n be that assumption, that for all k
less than or equal to n P of k.
And if you think about it for a moment,
just revising the induction hypothesis
to include that universal quantifier, for all k less than
or equal to n, means that the strong induction on P of k
becomes an ordinary induction on Q of n.
And we have a trivial change decorating
a bunch of occurrences of formulas
with for all we have converted and strong induction
into an ordinary induction.
So we see that strong induction and no power
above and beyond ordinary induction.
It just lets you omit a bunch of universal quantifiers
that would otherwise have to be made explicit if you were going
to do it by ordinary induction.
Then why use strong?
Just precisely, because it's cleaner.
You don't have to write those for all k less than
or equal to ends all over.
And now we come to the final question about,
what's the relation between the well-ordering principle
in induction?
Well, it's basically the same deal.
You can easily rephrase an induction proof.
An induction proof, just transform it's
template to fit the template of a well-ordering proof and vice
versa.
We're not going into the details of exactly how,
because it's not important, but it is routine.
It follows that well-ordering principle is not
adding any new power or even new perspective
on the mathematics of any given proof.
It's just a different way to organize and tell
the same story.
And it also means conceptually, which
is nice that these apparently different inference rules,
strong induction, ordinary induction,
well-ordering principle, there's really only one.
And the others can be justified in terms of it
and explained as variations of it.
So that's intellectually economical
to not have a proliferation of different reasoning
principles, which brings us to the question of which one
to use.
And all I can say is that it's a matter of taste.
The truth is that when I'm writing up proofs,
I will often try different versions.
I'll try it by ordinary induction.
And I'll try it by well-ordering.
And I'll read the two and decide which one seems
to come out the more cleanly.
And I'll go with that one.
So there isn't any simple rule about which to choose.
But in a certain sense, it really doesn't matter.
Just pick one.
The only exceptions to that, of course,
is when on an exam or similar setting,
you're told to use one of these particular methods
as a way to demonstrate that you understand it, then, of course,
you can't pick and choose.
So finally, we come to a pedagogical question about,
why is it that in 6042 we taught well-ordering and principal
first, in fact, the second lecture,
and are only now at the end of third week
getting to the induction principle, which is much more
familiar, and people argue they like it
better, at least most of them.
Well, the answer is it's a pedagogical strategy.
And it's one, in fact, which the authors disagree with, not
united on.
My view is that we're better off doing well-ordering principles
first.
And the reason is that our impression from conversations
with students, and surveys, and from exam performance
shows that only about 20% of the students
get induction no matter how hard we try to explain and teach it.
They report worrying about things
about that assuming P of n to prove p of n plus 1
is somehow circular.
And it's certainly measurable that 20% or so of the class
just can't reliably do proofs by induction.
Now this baffles the 80% to whom it's obvious
and who know how to do it easily.
And it baffles us instructors.
We can't figure out what the problem is that those 20% have.
And we've been trying to teach induction
lots of different ways.
On the other hand, nobody has trouble
believing the well-ordering principle and working with it.
And they certainly don't have any harder time using it
then they do using ordinary induction or strong induction.
And this conceptual problem about is it safe
and do I really believe in it just doesn't come up
with a well-ordering principle.
Everybody agrees that it's obvious
that a non-negative set of a non-empty set
of non-negative integers is going to have at least one.
And so we chose to do well-ordering right away,
because there's no overhead in explaining it.
And it lets us get going on interesting proofs
from the get go as opposed to waiting a while
or spending a couple of lectures working through induction
and get leaving that with the main if only method
that people have for proving things
about non-negative integers.