Math 5020 Handout 5
Basic Concepts of Axiomatic Set Theory (revised)
Throughout this handout we assume Extensionality, Pairing, Union, Power Set, Separation, Replacement,
and the axiom of
Existence
There exists a set.
1. The empty set
∅
exists and is unique.
2. A set
x
is
transitive
if
∪
x
⊆
x
.
3. A set
x
is an
ordinal
if
x
is transitive and
wellordered
by
∈
, i.e.,
∀
y
∈
x
(
y
̸∈
y
)
∀
y
∈
x
∀
z
∈
x
∀
u
∈
x
(
y
∈
z
∧
z
∈
u
→
y
∈
u
)
∀
y
∈
x
∀
z
∈
x
(
y
∈
z
∨
y
=
z
∨
z
∈
y
)
∀
w
∈ P
(
x
)
∃
y
∈
w
∀
z
∈
w
(
y
=
z
∨
y
∈
z
)
4.
(a)
∅
is an ordinal.
(b) If
α
is an ordinal and
β
∈
α
, then
β
is an ordinal.
(c) If
α, β
are ordinals and
α
⊂
β
, then
α
∈
β
.
(d) If
α, β
are ordinals then either
α
⊆
β
or
β
⊆
α
.
5. If
α, β
are ordinals, we define
α < β
⇐⇒
α
∈
β.
6. Define Ord to be the class of all ordinals. Ord is a proper class.
<
is a linear order on Ord.
7. If
α
is an ordinal, then so is
α
+ 1 :=
α
∪ {
α
}
.
α
+ 1 is called the
successor
of
α
. An ordinal
β
is a
successor ordinal
if
β
=
α
+ 1 for some ordinal
α
.
8. If
C
is a class of ordinals, then there is a unique ordinal inf
C
with the property that inf
C
∈
C
and
inf
C
≤
α
for all
α
∈
C
.
9. If
X
is a nonempty set of ordinals, then
∪
X
is an ordinal, and is in fact the least ordinal
β
with
α
≤
β
for all
α
∈
X
(thus it is also denoted as sup
X
).
10. If
α
is an ordinal but not a successor ordinal, then
α
= sup
{
β
:
β < α
}
=
∪
α
. In this case,
α
is called
a
limit ordinal
.
11. 0 :=
∅
is a limit ordinal. The ordinals
1 := 0 + 1
,
2 := 1 + 1
,
,
3 := 2 + 1
, . . .
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 Spring '11
 staff
 Set Theory, Mathematical logic, class a, Finite set

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