Download 6 Ordinals

6
Ordinals
In the past couple weeks, we’ve considered Cardinal numbers: numbers that
measure the size of a thing. Today, we’ll be taking a look at Ordinal numbers:
numbers that can count things.
The very first ordinal number is 0, which we’ll define as the empty set, {}.
After that, given any ordinal number ↵, we need a way to get ↵ + 1. To do this,
we’ll just define it as ↵ + 1 = ↵ [ {↵}.
Exercise 6.1
Explicitly write down what sets 0, 1, 2, 4 are.
This way of defining numbers has real upside: ↵ < if and only if ↵ 2
and only if ↵ ✓ . Further, every ordinal number is well-ordered:
Definition 9
least element.
if
An ordered set is well-ordered if every nonempty subset has a
For finite numbers, these properties are easy to ensure, but even infinite
ordinal numbers will satisfy them.
To reach infinite numbers, we need a di↵erent operation than adding 1; we
need something bigger than any finite number, and when you add 1 to something
finite, you’re still finite. So, define ! = 0 [ 1 [ 2 [ 3 [ 4 . . . = {0, 1, 2, 3, 4, . . .}.
You might recognize this as N; ! is our name for this set when we want to think
of it as an ordinal number.
All ordinals arise in one of these ways, either by adding 1 to another ordinal,
or by taking the union of a set of ordinals. Those that are defined in the first
way are called successor ordinals, while those only definable in the second way
are called limit ordinals.
6.1
Ordinal Arithmetic
Exercise 6.2
What is ! + 1?
We know how to add one; let’s use that to define more general addition.
Definition 10
For all ordinal numbers ↵,
• ↵+0=↵
• ↵ + ( + 1) = (↵ + ) + 1 for all ,
• If
=
0
Exercise 6.3
[
1
[
2
[ . . ., then ↵ +
= (↵ +
What are:
0 ) [ (↵ +
1 ) [ (↵ +
2) [ . . .
• !+3
• 5+!
• !+!
Now that we know how to add, we can use that to define multiplication:
Definition 11
For all ordinal numbers ↵,
• ↵·0=0
• ↵ · ( + 1) = (↵ · ) + ↵ for all ,
• If
=
0
Exercise 6.4
• !·2
• 3·!
• !·!
[
1
[
2
[ . . ., then ↵ ·
What are:
= (↵ ·
0)
[ (↵ ·
1)
[ (↵ ·
2)
[ ...
Now, just as multiplication is repeated addition, exponentiation can be defined as repeated multiplication.
Definition 12
For all ordinal numbers ↵,
• ↵0 = 1
• ↵
+1
• If
= ↵ · ↵ for all ,
=
0
Exercise 6.5
[
1
[
2
[ . . ., then ↵ = (↵ 0 ) [ (↵ 1 ) [ (↵ 2 ) [ . . .
What are:
• !2
• 8!
• !!
These are all the operations we’ll have for ordinals. Surprisingly, every
ordinal listed so far, when considered just as a set, is countable in size:
Exercise 6.6
Explain why the following are true:
1. ! + 1 is countable.
2. ! 2 is countable.
3. ! ! is countable (hint: try identifying it with the set of polynomials with
natural number coefficients).
Theorem 4 Transfinite Induction
Let C be a set of ordinal numbers, and assume
• 02C
• Whenever ↵ 2 C, ↵ + 1 2 C
• If ↵ is a limit ordinal, and for all
< ↵,
2 C, then ↵ 2 C.
Then C is the set of all ordinal numbers.
To prove this, consider the least ordinal ↵ 2
/ C, and use the above conditions
to see that ↵ must actually be in C.
Exercise 6.7
1. If
<
2. If ↵ <
3. If <
Prove the following:
then ↵ +
<↵+
then there exists a unique
and 0 < ↵, then ↵ ·
such that ↵ + = .
<↵· .
4. If ↵ > 0, and is arbitrary, then there exists a unique
⇢ < ↵ such that = ↵ · + ⇢.
5. If
<
and ↵ > 1 then ↵ < ↵ .
and a unique
Theorem 5 Cantor’s Normal Form Theory
Every ordinal ↵ > 0 can be represented uniquely in the form
↵ = ! 1 · k1 + . . . + ! n · kn ,
Where n
1, ↵
1 > ... >
n , and k1 , . . . , kn are nonzero natural
numbers.
Exercise 6.8
Prove Cantor’s Normal Form Theorem.
It is possible for the normal form to be ↵ = ! ↵ ; the smallest ordinal with
this property is called ✏0 . Surprisingly, even ✏0 is countable.
Exercise 6.9 Prove that there is no infinite sequence of ordinal numbers
↵0 , ↵1 , ↵2 , . . . such that ↵0 > ↵1 > ↵2 > . . . .
Exercise 6.10
Show that (! + 1) · 2 6= ! · 2 + 1 · 2 and (! · 2)2 6= ! 2 · 22 .
Exercise 6.11 Let ✏0 = ↵0 [ ↵1 [ ↵2 [ ↵3 [ . . . where ↵0 = ! and ↵n+1 = ! ↵n .
Show that ✏0 is the least ordinal ✏ such that ! ✏ = ✏.
Exercise 6.12
> ↵.
Show that for any ordinal ↵, there is a limit ordinal
with
Definition 13
↵ < and <
A limit ordinal > 0 is called indecomposable if there are no
such that ↵ + =
Exercise 6.13 Prove that a limit ordinal
if ↵ + = for all ↵ < .
> 0 is indecomposable if and only
Exercise 6.14 Prove that a limit ordinal
if = ! ↵ for some ↵.
> 0 is indecomposable if and only
Exercise 6.15
Prove that there exist uncountable ordinals.