14
I - Sets and Functions
first place {0}, the set whose only element is the empty set (box containing
an empty box), then {{0}}, the set whose only element is the set whose only
element is the empty set (box containing a box containing an empty box) etc.
The relation 0 E {{ {0}}} is false; the correct relation is 0 EEE {{ {0}}}: the
empty set belongs to a set which belongs to a set which belongs to {{ {0}}}.
These sets are pairwise distinct: the relation {{0}} = {{ {{0}}}} for example
would imply {0} = {{ {0}}} by the axiom of extension, then 0 = {{0}} for the
same reason, then {0} E 0, which is false. An empty box contains nothing,
not even an empty box.
This type of construction furnishes a possible definition of the primary
objects studied in mathematics, namely the whole numbers or natural integers. According to Zermelo, 1908, and in conformity with the programme of
reducing everything to set theory, one defines them by
(3.1)
0=0,
1 = {0},
2 = {{0}}, ... ;
simple abbreviations 18 for very particular sets. A computer would understand, but it would take twenty seconds for a machine running at 100 Mhz to
write or read the number 10 9 , assuming that one cycle is enough to recognise
the signs { and }.
Another method of defining the integers, equivalent 19 to the preceding,
due to von Neumann 20 , consists of putting
18 In particular the sign = used in these definitions is not that of set theory; when
introducing a definition the logicians (and now some mathematicians) prefer to
use the sign :=, the sign: warning the reader that one is introducing a definition
or new notation, and not a relation to be proved.
19 The precise mode of definition of the integers (or of any other mathematical
object) is of no importance so long as the various possible definitions lead to the
same theorems; the "nature" of mathematical objects is irrelevant because one
only asks them to be models of real objects. For the same reason the symbol
used by computer scientists to designate the number 15 is unimportant on the
theoretical plane, so long as the computers are programmed to recognise it.
20 1923; he was twenty years old and in the course of spending a few years learning
chemistry in Zurich because his father, a banker in Budapest, wanted to direct
him to a profession more lucrative than mathematics; he had already read Cantor
and Co. at least three years earlier. This definition of the integers, used by N.
Bourbaki, made certain French users of mathematics laugh at a certain period. A
list, even though incomplete, of von Neumann's activities, from 1937 - classical
explosives, operational research and game theory, the A-bomb, the H-bomb,
intercontinental missiles, computers - should reassure the philistines of his sense
of the concrete. But he was not intellectually cynical. Laurent Schwartz, Un
mathematicien aux prises avec le siecle (Odile Jacob, 1997), p. 288, translated
as A Mathematician Grappling with his Century (Birkhauser, 2001) writes about
me that I have "never pardoned von Neumann for having forsaken mathematics
to create computer science"; it is true that thanks to a colleague from Princeton
we knew vaguely in 1947, at Nancy, that von Neumann "was now doing numerical
calculations" though we were ignorant of their purpose; but, at least so far as
I am concerned, we know a lot more half a century later ... On von Neumann
I - Sets and Functions
first place {0}, the set whose only element is the empty set (box containing
an empty box), then {{0}}, the set whose only element is the set whose only
element is the empty set (box containing a box containing an empty box) etc.
The relation 0 E {{ {0}}} is false; the correct relation is 0 EEE {{ {0}}}: the
empty set belongs to a set which belongs to a set which belongs to {{ {0}}}.
These sets are pairwise distinct: the relation {{0}} = {{ {{0}}}} for example
would imply {0} = {{ {0}}} by the axiom of extension, then 0 = {{0}} for the
same reason, then {0} E 0, which is false. An empty box contains nothing,
not even an empty box.
This type of construction furnishes a possible definition of the primary
objects studied in mathematics, namely the whole numbers or natural integers. According to Zermelo, 1908, and in conformity with the programme of
reducing everything to set theory, one defines them by
(3.1)
0=0,
1 = {0},
2 = {{0}}, ... ;
simple abbreviations 18 for very particular sets. A computer would understand, but it would take twenty seconds for a machine running at 100 Mhz to
write or read the number 10 9 , assuming that one cycle is enough to recognise
the signs { and }.
Another method of defining the integers, equivalent 19 to the preceding,
due to von Neumann 20 , consists of putting
18 In particular the sign = used in these definitions is not that of set theory; when
introducing a definition the logicians (and now some mathematicians) prefer to
use the sign :=, the sign: warning the reader that one is introducing a definition
or new notation, and not a relation to be proved.
19 The precise mode of definition of the integers (or of any other mathematical
object) is of no importance so long as the various possible definitions lead to the
same theorems; the "nature" of mathematical objects is irrelevant because one
only asks them to be models of real objects. For the same reason the symbol
used by computer scientists to designate the number 15 is unimportant on the
theoretical plane, so long as the computers are programmed to recognise it.
20 1923; he was twenty years old and in the course of spending a few years learning
chemistry in Zurich because his father, a banker in Budapest, wanted to direct
him to a profession more lucrative than mathematics; he had already read Cantor
and Co. at least three years earlier. This definition of the integers, used by N.
Bourbaki, made certain French users of mathematics laugh at a certain period. A
list, even though incomplete, of von Neumann's activities, from 1937 - classical
explosives, operational research and game theory, the A-bomb, the H-bomb,
intercontinental missiles, computers - should reassure the philistines of his sense
of the concrete. But he was not intellectually cynical. Laurent Schwartz, Un
mathematicien aux prises avec le siecle (Odile Jacob, 1997), p. 288, translated
as A Mathematician Grappling with his Century (Birkhauser, 2001) writes about
me that I have "never pardoned von Neumann for having forsaken mathematics
to create computer science"; it is true that thanks to a colleague from Princeton
we knew vaguely in 1947, at Nancy, that von Neumann "was now doing numerical
calculations" though we were ignorant of their purpose; but, at least so far as
I am concerned, we know a lot more half a century later ... On von Neumann