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PROFESSOR: So let's
look at two examples

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of using the
well-ordering principle.

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One of them is pretty
obvious, and the other one

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is not hard but a little
bit more interesting.

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So what we're going to prove
is that every integer greater

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than one is a product of primes.

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So remember a prime is
an integer greater than 1

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that is only divisible by
itself and the number 1.

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It can't be expressed as
the product of other numbers

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greater than 1.

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So the way we're going to
prove this is by contradiction,

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and we're going to
begin by assuming

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suppose that there
were some numbers that

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were non-products of primes.

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OK.

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That is to say, the set of
non-products is non-empty.

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So applying the least--
the well-ordering principle

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to this non-empty
set of non-products,

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there's got to be a least one,
so m is a number greater than 1

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that is not a product of primes.

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Now, by convention, if
m itself was a prime,

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it's considered to be
a product of one prime,

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so we know that
m is not a prime.

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Now look.

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M is not a prime, or
if it was a prime,

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it would be a product
of just itself,

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so that means that it must
be a product of two numbers,

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call them j and k, where
j and k are greater than 1

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and less than m.

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That's what it means to
be a non-prime-- it's

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a product of j and k.

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Well, j and k are
less than m, so that

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means that they must be prime
products, because they're

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less than m and
greater than 1, and m

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is the smallest such number
that's not a product of primes.

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So we can assume that j is equal
to some product of prime say,

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p1 through p94, and k is
some other product of primes,

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q1 through q13, so you can
see where this is going.

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Now what we have is
that m, which is jk,

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is simply the
product of those p's

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followed by the product
of those q's and is,

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in fact, a prime product
which is a contradiction.

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So what did we assume that
led to the contradiction?

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We assumed that there were
some counter-examples,

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and there must not be any,
and no counter-examples

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means that, in fact, every
single integer greater than 1

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is indeed a product of
primes as [AUDIO OUT].

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Let's start looking at a
slightly more interesting

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example using the
well-ordered principle

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to reasoning about postage.

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So suppose that we
have a bunch of $0.05

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stamps and $0.03
stamps, and what

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I want to analyze is
what amounts of postage

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can you make out of $0.05
stamps and $0.03 stamps?

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So I'm going to introduce
a technical definition

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for convenience.

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Let's say that a
number n is postal.

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If I can make n plus
$0.08 postage from $0.03

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and $0.05 stamps.

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So this is what
I'm going to prove.

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I claim that every
number is postal.

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In other words, I can make
every amount of postage

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from $0.08 up.

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I'm going to prove
this by applying

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the well-ordering
principle, and as

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usual with
well-ordering principles

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we'll begin by
supposing that there was

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a number that wasn't postal.

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That would be a
counter-example, so if there's

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any number that's not
postal, then there's

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at least one m by the
well-ordering principal,

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because the set of
counter-examples

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is non-empty if some
number is not postal,

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so there's at least one.

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So what we know,
in other words, is

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that this least m that's
not postal has the property.

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It's not postal, and any
number less than it is postal.

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See what we can
figure out about m.

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First of all, m is not 0-- 0
is postal, because 0 plus $0.08

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can be made with a $0.03
stamp and a $0.05 stamp.

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M is not 0, because m is
supposed to be not postal.

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As a matter of fact,
by the same reasoning,

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m is not 1 because you can make
1 plus $0.08 with three $0.03,

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and m is not 2, because you
can make 2 plus $0.08-- $0.10--

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using two $0.05.

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So we've just figured out that
this least counter-example has

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to be greater than
or equal to 3,

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because 0, 1, and 2 are
not counter-examples.

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So we've got that
m is greater than

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or equal to 3, the
least non-postal number,

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so if I look at m
minus 3, that means

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it's a number that's
greater than or equal to 0,

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and it's less than
m, so it's postal,

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because m is the
least non-postal one.

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So, in other words, I can make--
out of $0.03 and $0.05 stamps,

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I can make m minus 3
plus $0.08 but, look,

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if I can make m
minus 3 plus $0.08,

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then obviously m is postal
also, because I just

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add $0.03 to that
and minus 3 number,

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and I wind up with
m plus $0.08, which

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says that m is postal
and is a contradiction.

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So assuming that there was
a least non-postal number,

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I reach a contradiction
and therefore there

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is no non-postal number.

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Every number is postal--
0 plus 8 is postal,

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1 plus 8 is postal,
2 plus 8 is postal.

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Every number greater
than or equal to $0.08

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can be made out of
$0.03 and $0.05 stamps.