18
I - Sets and Functions
X 2 =X xX,
X 3 = X X X X X, etc.
The existence of the cartesian product is plain from the naive point of view:
for the logicians it demands a proof. Now the elements z of X x Yare characterised by the following relation P {z }: there exist an x E X and ayE Y
such that z = {{x}, {x, yn. The existence of X x Y can therefore be deduced
from the axiom of separation so long as one knows in advance that those z
satisfying P{z} belong to a common set Z; but that precisely is the whole
problem.
Instead of just postulating the existence of X x Y axiomatically the logicians go much further. In the formula z = {{x}, {x,yn, z is a set whose
elements {x} and {x, y} are subsets of the union X U Y. The axiom which
allows one to resolve the problem affirms in a general way that, for every set
X, there exists a set P(X) whose elements are the subsets of x:
(4.4)
Y E P(X) {::::::} Y c X.
If, for example, X = {a, b, e} where a, b, e are pairwise distinct, P(X) has
the elements
0, {a}, {b}, {e}, {b,e}, {a,e}, {a,b}, {a,b,e}.
Returning to the cartesian product of X and Y, its elements z are, after
definition (2) of Kuratowski, sets whose two elements are subsets of Xu Y,
so elements of P(X U Y); one thus has z c P(X U Y) and so, by definition,
z E P(P(X U Y». The definition (2) of ordered pairs thus provides (axiom
of separation) a set
X x Y c P(P(XUY».
This argument (which you may well forget once you have understood it: the
sole thing to retain is the condition for two ordered pairs to be equal) may
appear somewhat esoteric and abstract, but it has the merit of showing that
the product X x Y can be constructed by means of the standard operations
of Set Theory, and for logicians, further, that it is logically founded, and obviates the risk of internal contradictions of the type of the Russell Paradox.
One has to appreciate that Logicians are an even more bizarre lot than Mathematicians: they feel the need to prove everything, including what is plain to
see to the "general public" . In matters electronical, these are content to press
the buttons on the black boxes and check that "it works"; the professionals
seek to understand what happens inside them.
The construction of P(X) for every set X is useful in many other circumstances. The definition of the real numbers proposed by Dedekind amounts
to saying, as we shall see at the beginning of Chapter II, that a real number
x is a set of rational numbers (intuitively, the set of ~ E Q such that ~ < x).
It is thus an element of P(Q) so, thanks to the axiom of separation one can
speak of the set IR C P(Q) of real numbers. After having constructed the real
I - Sets and Functions
X 2 =X xX,
X 3 = X X X X X, etc.
The existence of the cartesian product is plain from the naive point of view:
for the logicians it demands a proof. Now the elements z of X x Yare characterised by the following relation P {z }: there exist an x E X and ayE Y
such that z = {{x}, {x, yn. The existence of X x Y can therefore be deduced
from the axiom of separation so long as one knows in advance that those z
satisfying P{z} belong to a common set Z; but that precisely is the whole
problem.
Instead of just postulating the existence of X x Y axiomatically the logicians go much further. In the formula z = {{x}, {x,yn, z is a set whose
elements {x} and {x, y} are subsets of the union X U Y. The axiom which
allows one to resolve the problem affirms in a general way that, for every set
X, there exists a set P(X) whose elements are the subsets of x:
(4.4)
Y E P(X) {::::::} Y c X.
If, for example, X = {a, b, e} where a, b, e are pairwise distinct, P(X) has
the elements
0, {a}, {b}, {e}, {b,e}, {a,e}, {a,b}, {a,b,e}.
Returning to the cartesian product of X and Y, its elements z are, after
definition (2) of Kuratowski, sets whose two elements are subsets of Xu Y,
so elements of P(X U Y); one thus has z c P(X U Y) and so, by definition,
z E P(P(X U Y». The definition (2) of ordered pairs thus provides (axiom
of separation) a set
X x Y c P(P(XUY».
This argument (which you may well forget once you have understood it: the
sole thing to retain is the condition for two ordered pairs to be equal) may
appear somewhat esoteric and abstract, but it has the merit of showing that
the product X x Y can be constructed by means of the standard operations
of Set Theory, and for logicians, further, that it is logically founded, and obviates the risk of internal contradictions of the type of the Russell Paradox.
One has to appreciate that Logicians are an even more bizarre lot than Mathematicians: they feel the need to prove everything, including what is plain to
see to the "general public" . In matters electronical, these are content to press
the buttons on the black boxes and check that "it works"; the professionals
seek to understand what happens inside them.
The construction of P(X) for every set X is useful in many other circumstances. The definition of the real numbers proposed by Dedekind amounts
to saying, as we shall see at the beginning of Chapter II, that a real number
x is a set of rational numbers (intuitively, the set of ~ E Q such that ~ < x).
It is thus an element of P(Q) so, thanks to the axiom of separation one can
speak of the set IR C P(Q) of real numbers. After having constructed the real