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PROFESSOR: Let's
take a quick look

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at the axioms of Zermelo-Frankel
Set Theory With Choice.

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So the axioms of ZFC
define the standard theory

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of sets, which is now accepted
by most mathematicians

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as a reliable and simple basis
for developing and justifying

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all of mathematics.

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Among the axioms, maybe a
simple want to understand

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and really the motivation for
this short video is twofold.

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One is practice with
writing predicate formulas,

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and the other is to
think a little bit more

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about self application.

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So one of the basic
axioms of set theory

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is called
extensionality, which is

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capturing the idea that a set
is determined by its members.

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So let's consider the
assertion that two sets x and y

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have the same elements,
which we could write

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as a predicate formula
in set theory as for all

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x, x is a member of y, if and
only if x is a member of z.

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Now we could use this is
a definition of equality.

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It's what we mean by
y and z are equal.

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But we don't really need
to even introduce equality

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as a basic part of the
language and add axioms

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about how it behaves.

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There's one axiom that
covers things adequately,

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and that is that if two sets
have the same members, then

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they are members
of the same sets.

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So if all the members of x
and y are the same, then x

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and y are members of exactly
the same thing, which

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we could say this
way, for every x,

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y is an x, if and
only if z is an x.

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So that is one of the basic
axioms of Set Theory, maybe

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the starting one.

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Another one is the
Power Set axiom,

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which simply says that
every set has a power set.

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How would you say that in
the language of predicate set

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theory?

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Well, you'd say
that for every x,

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there is a p, which is going to
be the power set effects, such

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that for every set s, s is
a subset of x, if and only

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if s is a member of p.

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Remember, we know how to
express s as a subset of s

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in the language of
predicate calculus,

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mentioning only membership.

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So this is a good axiom
that says, yes, there

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is a set p consisting of
precisely the subsets of x.

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That set p called
the powers set of x.

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When you're trying to deal
with the Russell's paradox

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kind of issue, where you
define a set of element

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or a collection of
sets that satisfies

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some property, the safe
conservative version of saying

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that a set of elements that
satisfy some property really

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is a set, a collection
of elements that satisfy

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some property, really is a
set, the comprehension axiom's

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a simple version of an axiom
that allows you to do that.

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So basically, it
says that if s is

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a set and p of x is an arbitrary
predicate of set theory, which

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might in fact be one of
these dangerous things

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like x is not a member
of x, nevertheless,

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if you look at those
elements in the set

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S that satisfy P
of x, that's a set.

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In other words, the set of x and
s, such that P of x is a set,

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it means that any
definable collection

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of elements within a set
also form a proper subset.

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And the reason why
this matters is,

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remember, if I just talked
about not the set of x

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in a particular set
s the satisfied P

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of x, if I just talked about
the collection of x's that

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satisfied P of x, that's when
I start getting into Russell's

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paradox areas, when I
declare that the set of x

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such that P of x is a set
for unrestricted P of x.

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But all I get to do is put
a bound on the elements

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that x ranges over, that x is a
member of some particular set.

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Then it's safe to take all of
those x's that satisfy P of x.

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Now another particularly
interesting axiom

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of ZF which addresses this issue
of self membership and self

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reference is that the intuitive
idea that the elements of a set

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have to come before
the set itself.

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They have to be simpler
than the set itself,

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if you think about
sort of building up

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a set from successively
simpler elements to more

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complicated ones.

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In particular, you can't have
a set be a member of itself

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because then it's
not being built

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from things that are
simpler than it is

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or that came before it.

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In fact, you can't
even have a set that's

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a member of a member of itself.

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All of that kind of indirect
membership is forbidden.

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Now, how do you say
that is a nice axiom?

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Well, there's a very
elegant way to do it,

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and that is to say
that all sets are well

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founded under membership,
which means that you can't find

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an infinite sequence of
sets where each one has

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the next one as a member.

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Let's give a precise
way to formulate that.

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It's also good practice with
the formulas of set theory.

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Let me say that x is membership
minimal, epsilon minimal, in y

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means that x is a member of
y, but there's no element of x

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that's also in y.

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In other words, x is built out
of things that are not in y,

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but x itself is in y.

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So x kind of comes before any
of the other elements in y.

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It's built out of non-y stuff.

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So to say this with a formula we
could just say that x is in y,

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and for every z, if it's
in x, then it's not in y.

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So that's the definition of
x is membership minimal in y.

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And then the basic axiom of ZF,
called the Foundation Axiom,

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simply says that every
nonempty set has a membership

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minimal element.

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This is actually a
kind of generalization

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of the well ordering
principle that

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says that every nonempty
set of non-negative integers

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has a least element.

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This is a direct analogy.

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Just as the in
principle for integers

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implies that you can't
have an infinite decreasing

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sequence of
non-negative integers,

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the Foundation Axiom
actually implies

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that you can't have an infinite
sequence of sets, each of which

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is a member of the previous one.

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Here is a formula that's
asserting Foundation.

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For every x, if x is not empty,
that implies that there is a y,

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such that y is
membership minimal in x.

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What is the Foundation
got to do with membership?

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Well, the Foundation
Axiom actually

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will very quickly let us
conclude that no set is

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a member of itself.

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How does that work?

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Well, suppose that you are
interested in some set,

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and you'd like to verify
that the set can't

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be a member of itself.

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Well, let R be the set
consisting of just this set S

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that you're interested in.

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R is the singleton S, its
only element in S. Well,

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R is not empty.

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And by the Foundation Axiom,
it must have a membership

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minimal element.

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Now suppose that
S isn't S. We're

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going to reach a contradiction.

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The claim is that R has no
membership minimal element,

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and that violates
the Foundation Axiom,

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so you can't have S is a member
of S. Why does this follow?

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Well, look, R is supposed
to have a membership

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minimal element.

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Well, R's only got one element.

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So if it's got any membership
element, it's got to be S.

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But S this can't be
membership minimal

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because S is in R,
which means that S

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has an element in R in it.

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So S is not R minimal.

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And the Foundation
Axiom then immediately

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implies that you can't have
S be a member of S. S is not

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membership minimal in
R. And this argument

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extends in a nice way to
member of a member and member

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of a member, and we'll throw
a feedback on one question

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about that at you shortly.

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So looking at the Foundation
Axiom and the conclusion

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that no set is a member of
itself, what we can immediately

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conclude is that, first of
all, the collection of all sets

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can't be a set because if
the collection of all sets

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was a set, then it would
be a member of itself,

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and that's forbidden
by the S can't

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be a member of S consequence
of the Foundation Axiom.

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The second thing it tells
us is remember the set W

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from Russell's paradox?

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W was the collection
of those sets which

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are not members of themselves.

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Well, now we've just figured
out that this is all sets

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because no set is
a member of itself.

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So the sets that are not
members of themselves

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is everything, and that's
why W is not a set and not

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a member of itself,
which explains finally

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how the Foundation Axiom
resolves the Russell paradox.