§1. Set Theory
37
and SO to Y, one has Yo C Xo by the definition of Yo. One finds finally that
Xo == Yo whenever X and Yare equipotent ordinals. If E is any set whatever,
. and if one chooses an ordinal X equipotent to E, the corresponding ordinal
Xo does not depend on the choice of X and is equipotent to E. We can thus
agree to define Card(E) = Xo; this is the standard-set in the class of sets
equipotent to E. Exercise. An ordinal X is a cardinal if and only if X C Y
for all ordinals Y equipotent to X.
These results lie on the edge of the abyss: one step more and you fall into
metaphysics, into mysticism or into contradictions, for example if you speak
of "the set of ordinals". Indeed, suppose that such a set existed, and give it
the inevitable name: Q. Let us show that Q is again an ordinal. Every x E Q
and every y E x being an ordinal by (6) and so an element of Q, one sees that
x E Q implies x C Q, whence (0 1). If now a and b are two distinct elements
of il, one has either a E b, or b E a by (9), whence (0 2). To conclude, it
remains to deduce that Q E Q. Exercise. The union of a set of ordinals is an
ordinal.
We gave above the definition of finite sets according to Dedekind: the
set X is finite if every injection X ~ X is bijective. Many others were
found after him, but it is not always easy to establish the equivalence of all
these definitions, so we shall not attempt to do so. The Pole Alfred Tarski
for example characterised the finite sets in 1924 as follows: every family (Xi)
of subsets of X should possess a minimal element, i.e. one not containing
any other Xi ("proof": choose an Xi whose number of elements is minimal).
N is not finite, for if one denotes by Xn the set of integers p ~ n, one has
Xo = N ::) Xl ::) X 2 ::) ••• with strict inclusions; if N were finite, one would
have Xn = X n +l = ... from a certain integer non.
By reason of (11), this would also suffice to characterise the finite ordinals;
for example: an ordinal X is finite if, for any nonempty ordinal Y eX,
including Y = X, there exists an ordinal Z such that Y = Z u { Z}. The set
X = N is not finite in this sense: the condition is satisfied for every Y strictly
contained in X (since Y is a von Neumann set), but not for N itself: if one
had N = Z U {Z}, one would have ZeN and Z =1= N, so Z would be an
element of N by property (6), i.e. a von Neumann set, so s(Z) = N also, and
one would be led to the relation N E N.
Finally we remark that Cantor and his successors had a lot of fun defining
more or less algebraic operations on the cardinals, analogous to those known
for the whole numbers; one obtains them starting from th~ operations of
set theory. For example, one defines Card(X) + Card(Y) = Card(X U Y)
taking the precaution of assuming X and Y disjoint, Card(X)Card(Y) =
Card(X x V), Card(X)card(Y) = Card(F) where F is the set of maps from
X into Y, etc. The formula Card(X)+l = Card(X) characterises the infinite
sets and entrances the mystics, though not the financiers, they who will never
37
and SO to Y, one has Yo C Xo by the definition of Yo. One finds finally that
Xo == Yo whenever X and Yare equipotent ordinals. If E is any set whatever,
. and if one chooses an ordinal X equipotent to E, the corresponding ordinal
Xo does not depend on the choice of X and is equipotent to E. We can thus
agree to define Card(E) = Xo; this is the standard-set in the class of sets
equipotent to E. Exercise. An ordinal X is a cardinal if and only if X C Y
for all ordinals Y equipotent to X.
These results lie on the edge of the abyss: one step more and you fall into
metaphysics, into mysticism or into contradictions, for example if you speak
of "the set of ordinals". Indeed, suppose that such a set existed, and give it
the inevitable name: Q. Let us show that Q is again an ordinal. Every x E Q
and every y E x being an ordinal by (6) and so an element of Q, one sees that
x E Q implies x C Q, whence (0 1). If now a and b are two distinct elements
of il, one has either a E b, or b E a by (9), whence (0 2). To conclude, it
remains to deduce that Q E Q. Exercise. The union of a set of ordinals is an
ordinal.
We gave above the definition of finite sets according to Dedekind: the
set X is finite if every injection X ~ X is bijective. Many others were
found after him, but it is not always easy to establish the equivalence of all
these definitions, so we shall not attempt to do so. The Pole Alfred Tarski
for example characterised the finite sets in 1924 as follows: every family (Xi)
of subsets of X should possess a minimal element, i.e. one not containing
any other Xi ("proof": choose an Xi whose number of elements is minimal).
N is not finite, for if one denotes by Xn the set of integers p ~ n, one has
Xo = N ::) Xl ::) X 2 ::) ••• with strict inclusions; if N were finite, one would
have Xn = X n +l = ... from a certain integer non.
By reason of (11), this would also suffice to characterise the finite ordinals;
for example: an ordinal X is finite if, for any nonempty ordinal Y eX,
including Y = X, there exists an ordinal Z such that Y = Z u { Z}. The set
X = N is not finite in this sense: the condition is satisfied for every Y strictly
contained in X (since Y is a von Neumann set), but not for N itself: if one
had N = Z U {Z}, one would have ZeN and Z =1= N, so Z would be an
element of N by property (6), i.e. a von Neumann set, so s(Z) = N also, and
one would be led to the relation N E N.
Finally we remark that Cantor and his successors had a lot of fun defining
more or less algebraic operations on the cardinals, analogous to those known
for the whole numbers; one obtains them starting from th~ operations of
set theory. For example, one defines Card(X) + Card(Y) = Card(X U Y)
taking the precaution of assuming X and Y disjoint, Card(X)Card(Y) =
Card(X x V), Card(X)card(Y) = Card(F) where F is the set of maps from
X into Y, etc. The formula Card(X)+l = Card(X) characterises the infinite
sets and entrances the mystics, though not the financiers, they who will never