36
I - Sets and Functions
Cantor confined himself to considering well ordered sets, i.e. sets endowed
with an order relation satisfying (i), (ii) and (iii), and considered two such
sets as equivalent when there exists a bijection of the first onto the second
which preserves the order of the elements, which is much more strict than
the relation of equipotencej he did not think of imposing the condition (iv),
but systematically associated to every element x of a well ordered set X the
set of y E X such that y < Xj this is not very different, since one can show
that, for any well ordered set E, there exist an ordinal X and a bijection f
of X onto E which transforms the inequalities in X into the inequalities in
Ej further, X and f are determined uniquely by the order relation on E.
The condition (iv) of von Neumann thus provides a perfectly determined
"standard" in each equivalence class of well ordered setsj his definition of the
whole numbers similarly provides, in the "class" of sets of fourteen elements, a
Standard-Set of Fourteen Elements, namely the number 14. The physicists do
this every day when they compare their folding metres to the iridioplatinum
Standard-Metre lodged at the National Bureau of Standards at Washington,
D.C.
One should not be surprised that these ideas, introduced by Cantor about
1895 in a quasi-philosophical and very obscure style, did not, at that time,
arouse unanimous enthusiasm among his colleaguesj but some adopted them
immediately and tried to clarify them and to put them on a solid basisj in
1900, at the International Congress of Mathematicians, David Hilbert, who
was, with Henri Poincare, one of the two greatest mathematicians of the time,
proposed to his colleagues his famous list of the most important and difficult
problems of the agej "to prove the continuum hypothesis" was onej and one
knows his opinion on set theory, the Paradise into which Cantor has enabled
us to enter and which we shall never leave (I quote from memory). These
eulogies did not prevent the unhappy Cantor from spending a large part of
his last twenty years in psychiatric establishments. All his formalisable ideas
have been adopted, but not much of the detail of his definitions and proofs
has been retained, conceived as they were in an age when one still lacked a
precise language, a convenient notation, and the strict logical discipline introduced by his ... successive successors.
The ordinals may be used to define the cardinals of nO 8 really as sets and
not just as simple symbols. One cannot use the ordinals themselves since two
different ordinals can well be equipotent, for example N and its successor.
But in a given ordinal X the set of ordinals X' C X equipotent to X has, by
(5), a least element Xo (a nonstandard notationj we remark in passing that
the logicians use Greek letters a, /3, etc. to denote ordinals, to distinguish
them from the Hebrew cardinals); Xo may be X itself, for example if X = No
If Y i=- X is an ordinal equipotent to X and if one supposes for example that
Y C X in accordance with (8), one has Yo eYe X and thus Xo C Yo by the
definition of Xo. But then one has Xo C Y and since Xo is equipotent to X,
I - Sets and Functions
Cantor confined himself to considering well ordered sets, i.e. sets endowed
with an order relation satisfying (i), (ii) and (iii), and considered two such
sets as equivalent when there exists a bijection of the first onto the second
which preserves the order of the elements, which is much more strict than
the relation of equipotencej he did not think of imposing the condition (iv),
but systematically associated to every element x of a well ordered set X the
set of y E X such that y < Xj this is not very different, since one can show
that, for any well ordered set E, there exist an ordinal X and a bijection f
of X onto E which transforms the inequalities in X into the inequalities in
Ej further, X and f are determined uniquely by the order relation on E.
The condition (iv) of von Neumann thus provides a perfectly determined
"standard" in each equivalence class of well ordered setsj his definition of the
whole numbers similarly provides, in the "class" of sets of fourteen elements, a
Standard-Set of Fourteen Elements, namely the number 14. The physicists do
this every day when they compare their folding metres to the iridioplatinum
Standard-Metre lodged at the National Bureau of Standards at Washington,
D.C.
One should not be surprised that these ideas, introduced by Cantor about
1895 in a quasi-philosophical and very obscure style, did not, at that time,
arouse unanimous enthusiasm among his colleaguesj but some adopted them
immediately and tried to clarify them and to put them on a solid basisj in
1900, at the International Congress of Mathematicians, David Hilbert, who
was, with Henri Poincare, one of the two greatest mathematicians of the time,
proposed to his colleagues his famous list of the most important and difficult
problems of the agej "to prove the continuum hypothesis" was onej and one
knows his opinion on set theory, the Paradise into which Cantor has enabled
us to enter and which we shall never leave (I quote from memory). These
eulogies did not prevent the unhappy Cantor from spending a large part of
his last twenty years in psychiatric establishments. All his formalisable ideas
have been adopted, but not much of the detail of his definitions and proofs
has been retained, conceived as they were in an age when one still lacked a
precise language, a convenient notation, and the strict logical discipline introduced by his ... successive successors.
The ordinals may be used to define the cardinals of nO 8 really as sets and
not just as simple symbols. One cannot use the ordinals themselves since two
different ordinals can well be equipotent, for example N and its successor.
But in a given ordinal X the set of ordinals X' C X equipotent to X has, by
(5), a least element Xo (a nonstandard notationj we remark in passing that
the logicians use Greek letters a, /3, etc. to denote ordinals, to distinguish
them from the Hebrew cardinals); Xo may be X itself, for example if X = No
If Y i=- X is an ordinal equipotent to X and if one supposes for example that
Y C X in accordance with (8), one has Yo eYe X and thus Xo C Yo by the
definition of Xo. But then one has Xo C Y and since Xo is equipotent to X,