§ 1. Set Theory
35
axiom of Archimedes of Chap. II. To "construct" such a well ordering one
.would have to establish a bijection f between JR and an ordinal X having
the power of the continuum. This amounts to writing the elements of JR "one
after the other" as one does traditionally for the whole numbers; one might
first choose a sequence of elements of JR, then an other sequence following
the first, then a third following the second, etc., and since the union of a
countable infinity of countable sets is again countable, one would not have
exhausted JR after having put these sequences "end to end"; one would have
to continue transfinitely, as Cantor put it. The general theory shows that
such an ordinal X and such a bijection exist in the sense of the logicians,
but since this result depends on the axiom of choice (and is equivalent to it),
it is out of the question to exhibit either X or f by means of a reasonably
explicit procedure. One might well think that if an order relation possessing
the miraculous property (iii) were easy to see on JR, mathematicians would
not have waited until the XX th century to discover it. No one will ever see it.
According to von Neumann, 1923, an ordinal is, by definition, a well
ordered set subject to a supplementary condition which, apparently, no one
had thought of before him:
(iv) every x E X is the set of y E X such that y < x.
For an ordinal defined by (0 1) and (0 2), the property (iv) is trivial:
since y < x is equivalent to y E x this means that x is the set of y E X such
that y E x. In von Neumann's definition, which employs neither (0 1) nor
(0 2), (iv) shows that every element of X is, as a set 32 , a subset of X and
thus that y E x is equivalent to y < x. Condition (0 1) follows from this, and
(0 2) is none other than the condition (ii) above since y E x is equivalent to
y < x. Von Neumann's definition is therefore equivalent to the one we have
given.
The fact that the successor of an ordinal is again an ordinal shows that
the sets
N, N U {N}, N U {N} U {N U {N}}, etc.
are ordinals. On forming the union of the unending sequence formed by N
and its successive successors, if one dares to put it so, one obtains a new
ordinal to which one can again apply this process etc. Note in passing that
all these sets are countable, so very modest, since there exist ordinals of all
possible powers if one believes assertion (11). What distinguishes one from
the other is not their cardinal, it is the order in which their elements are,
at least virtually, written. To be precise, consider two equipotent ordinals X
and Y and suppose that there exists a bijection f from X onto Y such that
the relation x' < x" implies f(x' ) < f(x"). Then X = Y (not an obvious
result).
32 We again recall that, even if it is not obvious, every mathematical object is a
set.
35
axiom of Archimedes of Chap. II. To "construct" such a well ordering one
.would have to establish a bijection f between JR and an ordinal X having
the power of the continuum. This amounts to writing the elements of JR "one
after the other" as one does traditionally for the whole numbers; one might
first choose a sequence of elements of JR, then an other sequence following
the first, then a third following the second, etc., and since the union of a
countable infinity of countable sets is again countable, one would not have
exhausted JR after having put these sequences "end to end"; one would have
to continue transfinitely, as Cantor put it. The general theory shows that
such an ordinal X and such a bijection exist in the sense of the logicians,
but since this result depends on the axiom of choice (and is equivalent to it),
it is out of the question to exhibit either X or f by means of a reasonably
explicit procedure. One might well think that if an order relation possessing
the miraculous property (iii) were easy to see on JR, mathematicians would
not have waited until the XX th century to discover it. No one will ever see it.
According to von Neumann, 1923, an ordinal is, by definition, a well
ordered set subject to a supplementary condition which, apparently, no one
had thought of before him:
(iv) every x E X is the set of y E X such that y < x.
For an ordinal defined by (0 1) and (0 2), the property (iv) is trivial:
since y < x is equivalent to y E x this means that x is the set of y E X such
that y E x. In von Neumann's definition, which employs neither (0 1) nor
(0 2), (iv) shows that every element of X is, as a set 32 , a subset of X and
thus that y E x is equivalent to y < x. Condition (0 1) follows from this, and
(0 2) is none other than the condition (ii) above since y E x is equivalent to
y < x. Von Neumann's definition is therefore equivalent to the one we have
given.
The fact that the successor of an ordinal is again an ordinal shows that
the sets
N, N U {N}, N U {N} U {N U {N}}, etc.
are ordinals. On forming the union of the unending sequence formed by N
and its successive successors, if one dares to put it so, one obtains a new
ordinal to which one can again apply this process etc. Note in passing that
all these sets are countable, so very modest, since there exist ordinals of all
possible powers if one believes assertion (11). What distinguishes one from
the other is not their cardinal, it is the order in which their elements are,
at least virtually, written. To be precise, consider two equipotent ordinals X
and Y and suppose that there exists a bijection f from X onto Y such that
the relation x' < x" implies f(x' ) < f(x"). Then X = Y (not an obvious
result).
32 We again recall that, even if it is not obvious, every mathematical object is a
set.