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PROFESSOR: So we come to the
part that a lot of students

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have been asking
about, but which

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in fact is entirely optional.

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So that if you care to skip
this little piece of video,

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you're welcome to.

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It's not going to appear
on any exam or anything.

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But people have
consistently asked

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how they choose
which method of proof

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to use among ordinary
induction or strong induction

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or well-ordering.

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And the answer is that it's
hard to tell them apart,

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because in an easy
technical sense,

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they're really all equivalent.

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So let's look at
them one by one.

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First of all, it's clear
that ordinary induction

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is a special case
of strong induction.

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In the ordinary
induction, you're

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allowed to assume only p of n.

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In strong, you can assign
everything from p of 0

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up to p of n to
prove p of n plus 1.

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But you don't have to use
all the extra assumptions.

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You could just use p of n so
that any ordinary induction

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can be seen as just a special
case of a strong induction.

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It would be a little misleading
to call it strong induction,

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but it is strong induction.

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So why bother with it?

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Well, the answers, basically,
it's an expository difference.

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It helps your reader to know
that the proof for n plus 1

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is only going to depend on n not
on the k's that are less than n

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as they would typically in a
genuine strong induction proof.

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Second, is some argument that an
ordinary induction going from n

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to n plus 1 is more intuitive
than strong induction that

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goes from anywhere less than
or equal to n up to n plus 1.

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I'm not sure that I
subscribe to that,

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but I've heard people
make that claim.

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All right.

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There's another
perspective, which

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is interesting and maybe
surprising, which is,

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why not always use
ordinary induction?

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Oh, wait a minute.

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How do you replace
strong induction

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with ordinary induction?

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Well it's easy.

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Suppose that you've
proved for all m P of m

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using strong induction with
induction hypothesis P of m,

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what have you done?

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Well, it's the same
base case whether you're

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using ordinary or strong.

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But in strong, you would
do an inductive step

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where you actually assumed not
just p of n, but P of for all k

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less than or equal to n.

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And then using all those
hypotheses about P of k,

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you prove P of n plus 1
in the strong induction.

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Well, how do you turn it
into an ordinary induction?

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Just let Q of n be that
assumption, that for all k

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less than or equal to n P of k.

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And if you think
about it for a moment,

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just revising the
induction hypothesis

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to include that universal
quantifier, for all k less than

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or equal to n, means that the
strong induction on P of k

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becomes an ordinary
induction on Q of n.

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And we have a trivial
change decorating

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a bunch of occurrences
of formulas

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with for all we have
converted and strong induction

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into an ordinary induction.

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So we see that strong
induction and no power

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above and beyond
ordinary induction.

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It just lets you omit a bunch
of universal quantifiers

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that would otherwise have to be
made explicit if you were going

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to do it by ordinary induction.

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Then why use strong?

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Just precisely,
because it's cleaner.

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You don't have to write
those for all k less than

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or equal to ends all over.

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And now we come to the
final question about,

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what's the relation between
the well-ordering principle

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in induction?

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Well, it's basically
the same deal.

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You can easily rephrase
an induction proof.

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An induction proof,
just transform it's

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template to fit the template of
a well-ordering proof and vice

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versa.

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We're not going into the
details of exactly how,

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because it's not important,
but it is routine.

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It follows that well-ordering
principle is not

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adding any new power
or even new perspective

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on the mathematics
of any given proof.

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It's just a different
way to organize and tell

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the same story.

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And it also means
conceptually, which

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is nice that these apparently
different inference rules,

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strong induction,
ordinary induction,

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well-ordering principle,
there's really only one.

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And the others can be
justified in terms of it

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and explained as
variations of it.

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So that's intellectually
economical

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to not have a proliferation
of different reasoning

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principles, which brings us
to the question of which one

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to use.

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And all I can say is that
it's a matter of taste.

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The truth is that when
I'm writing up proofs,

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I will often try
different versions.

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I'll try it by
ordinary induction.

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And I'll try it
by well-ordering.

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And I'll read the two and
decide which one seems

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to come out the more cleanly.

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And I'll go with that one.

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So there isn't any simple
rule about which to choose.

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But in a certain sense,
it really doesn't matter.

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Just pick one.

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The only exceptions
to that, of course,

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is when on an exam
or similar setting,

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you're told to use one of
these particular methods

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as a way to demonstrate that you
understand it, then, of course,

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you can't pick and choose.

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So finally, we come to a
pedagogical question about,

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why is it that in 6042 we taught
well-ordering and principal

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first, in fact,
the second lecture,

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and are only now at
the end of third week

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getting to the induction
principle, which is much more

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familiar, and people
argue they like it

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better, at least most of them.

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Well, the answer is it's
a pedagogical strategy.

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And it's one, in fact, which
the authors disagree with, not

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united on.

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My view is that we're better off
doing well-ordering principles

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first.

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And the reason is that our
impression from conversations

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with students, and surveys,
and from exam performance

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shows that only about
20% of the students

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get induction no matter how hard
we try to explain and teach it.

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They report worrying
about things

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about that assuming P of
n to prove p of n plus 1

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is somehow circular.

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And it's certainly measurable
that 20% or so of the class

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just can't reliably do
proofs by induction.

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Now this baffles the
80% to whom it's obvious

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and who know how
to do it easily.

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And it baffles us instructors.

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We can't figure out what the
problem is that those 20% have.

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And we've been trying
to teach induction

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lots of different ways.

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On the other hand,
nobody has trouble

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believing the well-ordering
principle and working with it.

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And they certainly don't
have any harder time using it

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then they do using ordinary
induction or strong induction.

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And this conceptual
problem about is it safe

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and do I really believe
in it just doesn't come up

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with a well-ordering principle.

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Everybody agrees
that it's obvious

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that a non-negative
set of a non-empty set

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of non-negative integers is
going to have at least one.

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And so we chose to do
well-ordering right away,

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because there's no
overhead in explaining it.

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And it lets us get going
on interesting proofs

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from the get go as
opposed to waiting a while

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or spending a couple of lectures
working through induction

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and get leaving that with
the main if only method

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that people have
for proving things

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about non-negative integers.