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Intuitively, we think of a set as an organization and an element belonging to a set as a member of the organization. Accordingly, it also
seems intuitively clear that
(1) when we collect all objects of a certain kind the collection forms
an organization, i.e., forms a set, and, moreover,
(2) it is natural that when we collect members of a certain kind of
an organization, the collection is a sub-organization, i.e., a subset.
Item (1) above was challenged by Bertrand Russell in 1901 if we
accept the natural item (2), i.e., we must be careful when we use the
word “all”:
(Russell Paradox) The collection of all sets is not a set!
Proof. Suppose the collection of all sets is a set S. Consider the subset
U of S that consists of all sets x with the property that each x does
not belong to x itself.
Now since U is a set, U belongs to S. Let us ask whether U belongs
to U itself.
Since U is the collection of all sets each of which does not belong to
itself, thus if U belongs to U , then as an element of the set U we must
have that U does not belong to U itself, a contradiction.
So, it must be that U does not belong to U . But then U belongs to
U itself by the very definition of U , another contradiction.
The absurdity results because we assume S is a set. Hence S is not
a set.
The paradox is in spirit parallel to the following statement of Russell:
The barber shaves all those who do not shave themselves.
The question is: What about the barber himself? If he shaves himself
then he does not shave himself. But if he does not shave himself he
shaves himself.
To overcome Russell’s paradox, an axiomatic set theory was proposed
in the 1920s, known as the Zermelo-Frankel theory, which introduces
nine axioms that form the foundation of mathematics.
To begin with, in the system we study, there are two predicates ∈,
read belongs to, and =, read equals. We also have a symbol ∅, read the
empty set, reserved for a fixed object. Otherwise, x, y, z, p, q, r, a, b, c,
etc., and their uppercase counterparts denote variable objects. The
predicate = has the property:
x = x for all x;
x = y implies y = x for all x, y;
x = y and y = z imply x = z for all x, y, z.
Definition 1. A is called a set if either there is an a ∈ A or else A = ∅.
If a ∈ A, then a is called an element of the set A. We say a set A is a
subset of a set B, written A ⊂ B or B ⊃ A, read A is contained in B
or B contains A, if a ∈ A implies a ∈ B for all a ∈ A.
We will use the notation
{a, b, c, · · · }
to denote a set when its elements are emphasized.
Axiom 1 (Axiom of Extensionality). Given sets A and B, then A = B
if and only if they have the same elements.
Axiom 2 (Axiom of Schema of Separation). Given a set A and a
statement (the term can be defined precisely in Mathematical Logic)
Ψ(x) of the elements x of A, there exists a set B such that it consists
of all x ∈ A that satisfy Ψ(x).
We will denote B in Axiom 2 by the symbol
B = {x ∈ A : Ψ(x)}
Question 1: Show that the empty set ∅ contains no elements at all,
as its name suggests.
(Hint: Consider
B = {x ∈ ∅ : x 6= x},
where the statement Ψ(x) is x 6= x. Axiom 2 says that B is a set. Now
apply Definition 1 to assert that B = ∅.)
Axiom 3 (Union Axiom). Given sets A and B, there is a set C that
contains exactly the elements of A and B. This set C is called the
“union” of A and B, denoted A ∪ B.
Note: Given sets A and B, Axiom 3 assures the existence of the set
A ∪ B. Then Axiom 2 implies the existence of the set
{x ∈ A ∪ B : x ∈ A and x ∈ B},
which we call the intersection of A and B, denoted A ∩ B.
Axiom 4 (Pairing Axiom). Given sets A and B, there is a set C whose
elements are exactly the two sets. That is,
C = {A, B}.
Note: For any set B, Axiom 4 implies the existence of the set {B, B},
which we denote by {B} for short.
Axiom 5 (Sum Axiom). Given a set A, there is a set C that is the
collection of elements of the set elements of A.
For example, for
A = {{1}, {1, 2}, {4, 5}, Mt. Everest},
we have C = {1, 2, 4, 5}.
Axiom 6 (Power Set Axiom). Given a set A, there is a set C whose
elements are all the subsets of A.
For example, for A = {1, 2, 3}, we have
C = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.
Question 2: Show that the empty set ∅ is always a subset of any set.
Axiom 7 (Axiom of Regularity). Given a set A 6= ∅, there is an x ∈ A
such that when x is a set it satisfies A ∩ x = ∅.
For example, for
A = {{1}, {1, 2}, 3, 4},
we can choose x = {1}.
Axiom of Regularity is devised to assure that A ∈
/ A for any set A,
as our intuition expects.
Question 3: Show that A ∈
/ A holds for any set A.
(Hint: Consider the set {A, {A}}; why is it a set in the first place?)
Axiom 8 (Axiom of Infinity). There is a set A such that it contains
the empty set ∅, and if a set a is in A, then its successor a0 := a ∪ {a}
is also in A.
This axiom has to do with the notion of natural numbers, as we will
see later.
Axiom 9 (Axiom of Choice). Given sets I and A, for each i ∈ I let
Ai be a subset of A (i.e., we are given subsets of A indexed by elements
of I). Then there is a set C formed by picking one element from each
Ai .
Axiom 9 can be made more precise when we define the notion of a
function later.
In this axiomatic system of sets, one is only allowed to make assertions by logical inferences, starting from the axioms. The conclusion of
an assertion is called a theorem (in particular, an axiom is a theorem
without doing any logical inferences). For example, the statement in
Question 1 is a theorem. One can freely quote theorems one has proven
thus far to arrive at new theorems, etc.
However, the most important issue in an axiomatic system is its
consistency. That is, one has to make sure that the axioms do not have
intrinsic logical contradiction among themselves.
Kurt Gödel in the 1930s proved that if one assumes that the first
eight axioms of set theory above are consistent, i.e., have no contradiction among them, then the ninth axiom, the Axiom of Choice, is
consistent with the previous eight! But then in the 1960s, Paul Cohen proved that if the first eight axioms of set theory are consistent,
then the negation of the Axiom of Choice is also consistent with the
previous eight! The conclusion of these two stunning theorems, from
which Gödel and Cohen won the Fields Medal, is therefore that the
Axiom of Choice is independent of the first eight axioms, i.e., it cannot
be logically derived from the first eight, nor can its negation. It simply has nothing to do with the first eight axioms. Hence, whether we
adopt the Axiom of Choice is a matter of taste! Dropping the Axiom
of Choice would lead to a different mathematics from ours. This new
mathematics would be equally valid as ours, in which we adopt the
Whether the first eight axioms are consistent is unknown.
Gödel also proved yet another startling theorem, which is named
his incompleteness theorem. It states, roughly, that in an axiomatic
system (the term can be made precise in Mathematical Logic), if it
contains natural numbers and if the system is consistent, then there
must be a true statement that cannot be logically deduced from the
axioms! Equivalently, if all true statements can be logically deduced
from the axioms, then the system is inconsistent! In particular, our
mathematical system is incomplete!
For a beautiful exposition of this incompleteness theorem, you can
read the Pulitzer Prize winner
Gödel, Escher, Bach: an Eternal Golden Braid, 20th anniversary ed.
by Douglas R. Hofstadter.