32
I - Sets and Functions
or b E a and that the relation a E b is equivalent to {a C b and a i= b}. If a,
b, c are three elements of X and both a E band b E c, then a E c. Finally
one remarks that if X and Yare two integers defined Ii La von Neumann,
the relation X ::; Y, which everyone knows, becomes X C Y, showing that
inequalities between integers are reducible to membership relations or set
inclusions. One also sees that if X and Y are two von Neumann integers, one
always has X c Y or Y eX.
Starting from these properties of von Neumann sets (or whole numbers),
one can generalise by calling an ordinal (number or set) any set X possessing
the following properties :
(0 1) the relation x E X implies x C X;
(02) for any a,b E X, one has either a E b, or a = b, or b E a.
The simplest ordinal is naturally 0, but there are others, to start with the
sets of von Neumann, who gave a slightly different definition of the ordinals,
though it is equivalent to the one above.
The set N is also an ordinal. To check (0 1), one remarks that every x E N
is a von Neumann integer, from which all its elements are too, so belong to
N, whence x c N. To check (0 2), one notes that any von Neumann integer
is an element of all its successors and that if b i= a is not one of the successors
of a, then a is one of the successors of b (one need only be able to read and
count ... ).
These sets possess rather amusing properties; as in Euclidean geometry
and even for the same reason - one constructs an entirely autonomous theory
knowing hardly anything -, they are proved in a technically elementary way
by applying the definitions.
First note that (0 1) can be written in the form
Xc P(X)
or in the form
x EE X ==} x E X
since an element of a element of X is also an element of a subset of X, and
so of X.
Furthermore, the three cases in (02) are pairwise exclusive; for a E band
a = b would imply a E a, while a E band b E a is, like a E a, forbidden by
the providential axiom of regularity at the end of nO 2.
(1) Every intersection of ordinals is an ordinal. Clear from the definition.
(2) If X is an ordinal, then s(X) = X U {X} is an ordinal. Verification of
(0 1): x E s(X) implies either x E X, whence x c Xc s(X), or x = X and
again x C s(X). Verification of (0 2): if a, b E s(X), one has either a E X
and b EX, whence a E b or b = a or b E a since X is an ordinal, or a E X
and b E {X}, whence b = X and thus a E b, or a, b E {X}, whence a = b,
qed.
I - Sets and Functions
or b E a and that the relation a E b is equivalent to {a C b and a i= b}. If a,
b, c are three elements of X and both a E band b E c, then a E c. Finally
one remarks that if X and Yare two integers defined Ii La von Neumann,
the relation X ::; Y, which everyone knows, becomes X C Y, showing that
inequalities between integers are reducible to membership relations or set
inclusions. One also sees that if X and Y are two von Neumann integers, one
always has X c Y or Y eX.
Starting from these properties of von Neumann sets (or whole numbers),
one can generalise by calling an ordinal (number or set) any set X possessing
the following properties :
(0 1) the relation x E X implies x C X;
(02) for any a,b E X, one has either a E b, or a = b, or b E a.
The simplest ordinal is naturally 0, but there are others, to start with the
sets of von Neumann, who gave a slightly different definition of the ordinals,
though it is equivalent to the one above.
The set N is also an ordinal. To check (0 1), one remarks that every x E N
is a von Neumann integer, from which all its elements are too, so belong to
N, whence x c N. To check (0 2), one notes that any von Neumann integer
is an element of all its successors and that if b i= a is not one of the successors
of a, then a is one of the successors of b (one need only be able to read and
count ... ).
These sets possess rather amusing properties; as in Euclidean geometry
and even for the same reason - one constructs an entirely autonomous theory
knowing hardly anything -, they are proved in a technically elementary way
by applying the definitions.
First note that (0 1) can be written in the form
Xc P(X)
or in the form
x EE X ==} x E X
since an element of a element of X is also an element of a subset of X, and
so of X.
Furthermore, the three cases in (02) are pairwise exclusive; for a E band
a = b would imply a E a, while a E band b E a is, like a E a, forbidden by
the providential axiom of regularity at the end of nO 2.
(1) Every intersection of ordinals is an ordinal. Clear from the definition.
(2) If X is an ordinal, then s(X) = X U {X} is an ordinal. Verification of
(0 1): x E s(X) implies either x E X, whence x c Xc s(X), or x = X and
again x C s(X). Verification of (0 2): if a, b E s(X), one has either a E X
and b EX, whence a E b or b = a or b E a since X is an ordinal, or a E X
and b E {X}, whence b = X and thus a E b, or a, b E {X}, whence a = b,
qed.