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Econ 204
Set Formation and the Axiom of
Choice
In this supplement, we discuss the rules underlying set formation and the
Axiom of Choice.
We generally begin with a set of elements, such as the natural numbers
N
, the rational numbers
Q
,therea
lnumbers
R
, or an abstract set like the
set
X
of all points of an unspeciFed metric space.
Given any set
X
,w
ecanfo
rm2
X
, often called the
power set
of
X
;2
X
is the set of all subsets of
X
. Thus, we can form the set
N
of all natural
numbers, 2
N
, the set of all subsets of
N
;
∅
,
{
1
,
2
}
,
{
2
,
4
,
6
,...
}
are elements
of 2
N
.
We can also form 2
2
N
=2
(
2
N
)
, the set of all subsets of the set of all
subsets of the natural numbers.
An element of 2
2
N
is a set of subsets of
the natural numbers; for example,
{∅}
,
{∅
,
N
}
,
{{
1
}
,
{
2
}
,
{
2
,
4
,
6
,...
}}
and
{{
2
}
,
{
4
}
,
{
6
}
,...
}
are elements of 2
2
N
.
Let
X
be any set, and
P
(
x
) a mathematical statement about a variable
x
.Then
{
x
∈
X
:
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 Summer '08
 ANDERSON

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