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Math 5020 Handout 6
Ordinal Arithmetic
From now on we assume ZF
−
Regularity, until further notice.
1. (Transﬁnite Induction) Let
C
be a class of ordinals and assume that
(i) 0
∈
C
;
(ii) if
α
∈
C
then
α
+ 1
∈
C
;
(iii) if
α
is a nonzero limit ordinal and
β
∈
C
for all
β < α
, then
α
∈
C
.
Then
C
= Ord.
2. A
sequence
is a function with domain Ord. We denote a sequence by
⟨
a
α
:
α
∈
Ord
⟩
. If
α
is an ordinal,
a
sequence of length
α
is function with domain
α
. A
ﬁnite sequence
is a sequence of length
n
for some
ﬁnite ordinal
n
∈
ω
.
3. (Transﬁnite Recursion) For every function
G
there is a unique function
F
such that for all ordinals
α
,
F
(
α
) =
G
(
F

α
).
4. Let
W
be a set and
<
⊆
W
×
W
.
<
is a
wellorder
if it is a linear order and any nonempty subset of
W
has a
<
least element.
5. Every wellorder is isomorphic to a unique ordinal. I.e., if
W
is a set and
<
is a wellorder on
W
, then
there is a unique ordinal
α
such that there exists a bijection
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 Spring '11
 staff
 Math, Set Theory

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