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PROFESSOR: The well
ordering principle

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is one of those
facts in mathematics

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that's so obvious that
you hardly notice it.

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And the objective of
this brief introduction

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is to call your attention to it.

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We've actually used it already.

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And in subsequent segments
of this presentation,

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I'll show lots of
applications of it.

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So here's a statement of
the well ordering principle.

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Every nonempty set of
nonnegative integers

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has a least element.

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Now this is probably familiar.

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Maybe you haven't
even thought about it.

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But now that I mentioned it,
I expect it's a familiar idea.

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And it's pretty
obvious too if you

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think about it for a minute.

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Here's a way to think about it.

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Given an nonempty set of
integers, you could ask,

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is 0 the least element in it?

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Well, if it is,
then you're done.

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Then you could say, is 1
the least element in it?

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And if it is, you're done.

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And if it isn't, you could
say 2, is 2 the least element?

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And so on.

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Given that it's not
empty, eventually you're

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to hit the least element.

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So, if it wasn't
obvious before, there

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is something of a
hand-waving proof of it.

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But I want to get you
to think about this well

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ordering principle a little bit
because it's not-- there are

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some technical parts
of it that matter.

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So for example, suppose I
replace nonnegative integers

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by nonnegative rationals.

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And I asked does every nonempty
set of nonnegative rationals

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have a least element?

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Well, there is a least
nonnegative rational, namely 0.

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But not every nonnegative set of
rationals has a least element.

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I'll let you think
of an example.

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Another variant is
when, instead of talking

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about the nonnegative
integers, I just

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talk about all the integers.

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Is there a least integer?

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Well, no, obviously because
minus 1 is not the least.

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And minus 2 is not the least.

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And there isn't
any least integer.

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We take for granted the
well ordering principle just

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all the time.

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If I ask you, what was the
youngest age of an MIT graduate

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well, you wouldn't for a
moment wonder whether there

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was a youngest age.

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And if I asked you for the
smallest number of neurons

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in any animal, you
wouldn't wonder

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whether there was or wasn't
a smallest number of neurons.

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We may not know what it is.

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But there's surely a
smallest number of neurons

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because neurons are
nonnegative integers.

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And finally, if I ask you what
was the smallest number of US

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coins that could
make $1.17, again, we

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don't have to worry
about existence

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because the well ordering
principle knocks that

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off immediately.

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Now for the remainder
of this talk,

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I'm going to be talking about
the nonnegative integers

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always, unless I
explicitly say otherwise.

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So I'm just going to
is the word number

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to mean nonnegative integer.

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There's a standard
mathematical symbol

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that we use to denote
the nonnegative integers.

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It's that letter N at
the top of the slide

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with a with a
diagonal double bar.

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These are sometimes called
the natural numbers.

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But I've never been able
to understand or figure out

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whether 0 is natural or not.

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So we don't use that phrase.

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Zero is included in N,
the nonnegative integers.

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And that's what we call
them in this class.

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Now, I want to point
out that we've actually

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used the well ordering
principle already

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without maybe not
noticing it, even

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in the proof that the square
root of 2 was not rational.

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That proof began by saying,
suppose the square root of 2

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was rational, that is, it was a
quotient of integers m over n.

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And the remark was that you
can always express a fraction

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like that in lowest terms.

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More precisely, you can always
find positive numbers m and n

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without common factors, such
that the square root of 2

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equals m over n.

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If there's any fraction equal
to the square root of 2,

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then there is a lowest
terms fraction m

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over n with no common factors.

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So now we can use well
ordering to come up

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with a simple, and hopefully
very clear and convincing,

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argument for why
every fraction can

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be expressed in lowest terms.

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In particular, let's
look at numbers m and n

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such that the square root
of 2 is equal to m over n--

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that fraction.

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And let's just choose the
smallest numerator that works.

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Find the smallest numerator
m, such that squared of 2

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is equal to m over n.

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Well, I claim that
that fraction, which

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uses the smallest
possible numerator,

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has got to be in lowest
terms because suppose

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that m and n had a common
factor c that was greater

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than 1-- a real common factor.

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Then you could replace
m over n by m over c,

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the numerator is a
smaller numerator that's

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still an integer, and n over c.

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The denominator is
still an integer.

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And we have a numerator
that's smaller than m

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contradicting the way that we
chose m in the first place.

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And this contradiction,
of course,

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implies that m and n
have no common factors.

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And therefore, as claimed,
m over n is in lowest terms.

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And of course, the
way I formulated

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this was for our application
of the fraction that was

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equal to the square root of 2.

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But this proof actually shows
that any rational number,

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any fraction, can be
expressed in lowest terms.