y
z
(z
x
z
y
.
We start with A Brief History of Set Theory
Axiom ZF1 - Sets with the same members are equal - (Extensionality).
If x and y are sets then x
y
z
(zx
zy
.
x
y
x
y
z
(z
x
z
y
Axiom ZF2 - The "Empty Set" is a set. We write it as
There exists a set
such that
x(
(x
)).
y
x(
(x
y))
Note by ZF1
is unique.
Axiom ZF3 - Set Formation - Unordered Pairs.
If x,y are sets then there exists a set z such that
a
a
z
a
x or a
y
x
y
z
a
a
z
a
x or a
y
.
Notation: We write this unique set as {x,y} . Note that {x,x} which is the same as {x} is not the set x.
Axiom ZF4 - Set Formation - Union.
If x is a set then there exists a set y such that
a
y
a
y
z
z
x
and
a
z
.
Notation: We
write y
x
, or some variant thereof
Axiom ZF5 - Set Formation - Power set.
If x is a set then there exists a set z such that
y
y
z
y
x
.
x
z
y
y
z
a
(a
y
a
x
) )
Notation:
a
(a
y
a
x
) is just a definition of
inclusion y
x
Axiom ZF6 - Set Formation - Selection.
If z and
x
is a proposition then there exits a set y such that
x(x
y
x
z
and
x
.
z
y
x
x
y
x
z
and
x
.
Note that this avoids Russell's Paradox since we require x to be a member of a "known Set."
Notation:We write
y
x
z
x
.
Axiom ZF7 - There is an infinite Set
There exists a Set m such that
m
and
x
x
m
x
x
m
.
m
m
and
x
x
m
x
x
m
Translation:
x
{x}
is the Set containing all the members of x and the Set
x itself.
A Computation:
Step:0
m
This Set has zero members.
1
m
Since
,this set has one member, the Set with 0 members.
2
m
This set has two members, the Set with 0 members, a Set with 1 member.
3
m
This set has three members, the Set with 0 members, a Set with 1 member and a Set with 2 members.
.
.
n
ZF7 allows us to "construct" the natural numbers.
Axiom ZF8 - Replacement( Functional Image)
Let
x
y
be
a proposition.
Suppose 
x
y
y
x
y
and
x
y
y
y
That is,
x
y
is a "single valued function."
then
a
b
c 
c
b
d
d
a
and
d
c
Translation: The functional image of a Set is a Set.
x
y
y
x
y
and
x
y
y
y
a
b
c 
c
b
d
d
a
and
d
c
Axiom ZF9 - There are not Russell's Paradox like Sets (Regularity).
If x is a set and x
then
y
y
x
and
a
a
x
and
a
y
.
x
x
y
y
x
and
a
a
x
and
a
y
Translation: Every non-empty set contains a member that does not any members in common with it.
Note
a(
(a
x
and
a
y)).
can be read
x
y
.
This can be shown to rule out the Russell set.