I - Sets and Functions
5
the axioms starting from which one can erect the whole of mathematics in a
perfectly coherent way, without, it is to be hoped, stumbling upon an internal
contradiction. The rules imposed, while allowing all the standard mathematical arguments and objects, are strict enough for the early paradoxes to be, one
hopes, impossible to formulate without infringements; to wit, safety barriers
beyond which one ventures at one's own risk and peril. This domain, which
has been the subject of a large number of works for a century, still presents
very difficult problems and has sometimes given rise to intense debate; most
mathematicians observe this from a distance, content with a "naive", i.e. not
strictly formalised version, of Logic and Set Theory.
The principal result of this work is that every mathematical object can be
considered as a set and indeed that the only logically correct way to define a
mathematical object is to say that it is composed of one or more sets subject
to explicitly stated conditions. This method suffers from the inconvenience
of making the seemingly simplest mathematical objects, the real numbers for
example, appear as extremely complex elaborations of sets; in the definition
of the real numbers as "cuts", due to Dedekind, and simplified by Peano
and Russell, which we shall give at the beginning of the next chapter, a real
number, for example rr, is by definition a set of rational numbers (naively: the
set of all rational numbers < rr); a rational number, in its turn, is 5 the set of
all pairs of integers (p, q) of arbitrary sign such that x = p/q, q =f. 0; an integer
n of arbitrary sign is itself the set of all pairs (p, q) of whole numbers such
that p - q = n; a whole number, at last, is a set, the number 3 for example
being a set of three elements chosen once and for all; thus the number rr
becomes a set of sets of sets of sets, all infinite except for the last. One might
therefore, starting from the whole numbers, establish a kind of hiemrchy of
complexity in the universe of mathematical objects, as do certain logicians.
But of course from the moment these objects and the operations to which
they may be subjected have been defined by applying the standard operations
of set theory to objects already known, one forgets their explicit definitions,
their extreme complexity making manipulating them quite impossible; one
confines oneself to arguing as was always done, up to one detail: one knows
exactly, or one could know, if one put one's mind to it, what the symbolrr
signifies, as it leaves metaphysics and enters into mathematics 6 .
Following this historic evolution one can contemplate Set Theory from two
different points of view: on the one hand from the "naive" point of view of
5 See for example §§5 and 28 of my Cours d'algebre (Hermann, 1966) or Algebra
(Addison-Wesley. 1972)
6 The physicist Emilio Segre explains in his memoirs that, when he was in highschool, he did not understand why his mathematics teacher stressed the need to
define the real numbers by means of Dedekind sections, the concept of number
seeming to be sui generis. This only proves that a Nobel Laureate in physics
may not understand the mathematics he has been using during his whole life, or,
if one prefers, not understand the difference between Physics and Mathematics.
See the beginning of Chapter II.
5
the axioms starting from which one can erect the whole of mathematics in a
perfectly coherent way, without, it is to be hoped, stumbling upon an internal
contradiction. The rules imposed, while allowing all the standard mathematical arguments and objects, are strict enough for the early paradoxes to be, one
hopes, impossible to formulate without infringements; to wit, safety barriers
beyond which one ventures at one's own risk and peril. This domain, which
has been the subject of a large number of works for a century, still presents
very difficult problems and has sometimes given rise to intense debate; most
mathematicians observe this from a distance, content with a "naive", i.e. not
strictly formalised version, of Logic and Set Theory.
The principal result of this work is that every mathematical object can be
considered as a set and indeed that the only logically correct way to define a
mathematical object is to say that it is composed of one or more sets subject
to explicitly stated conditions. This method suffers from the inconvenience
of making the seemingly simplest mathematical objects, the real numbers for
example, appear as extremely complex elaborations of sets; in the definition
of the real numbers as "cuts", due to Dedekind, and simplified by Peano
and Russell, which we shall give at the beginning of the next chapter, a real
number, for example rr, is by definition a set of rational numbers (naively: the
set of all rational numbers < rr); a rational number, in its turn, is 5 the set of
all pairs of integers (p, q) of arbitrary sign such that x = p/q, q =f. 0; an integer
n of arbitrary sign is itself the set of all pairs (p, q) of whole numbers such
that p - q = n; a whole number, at last, is a set, the number 3 for example
being a set of three elements chosen once and for all; thus the number rr
becomes a set of sets of sets of sets, all infinite except for the last. One might
therefore, starting from the whole numbers, establish a kind of hiemrchy of
complexity in the universe of mathematical objects, as do certain logicians.
But of course from the moment these objects and the operations to which
they may be subjected have been defined by applying the standard operations
of set theory to objects already known, one forgets their explicit definitions,
their extreme complexity making manipulating them quite impossible; one
confines oneself to arguing as was always done, up to one detail: one knows
exactly, or one could know, if one put one's mind to it, what the symbolrr
signifies, as it leaves metaphysics and enters into mathematics 6 .
Following this historic evolution one can contemplate Set Theory from two
different points of view: on the one hand from the "naive" point of view of
5 See for example §§5 and 28 of my Cours d'algebre (Hermann, 1966) or Algebra
(Addison-Wesley. 1972)
6 The physicist Emilio Segre explains in his memoirs that, when he was in highschool, he did not understand why his mathematics teacher stressed the need to
define the real numbers by means of Dedekind sections, the concept of number
seeming to be sui generis. This only proves that a Nobel Laureate in physics
may not understand the mathematics he has been using during his whole life, or,
if one prefers, not understand the difference between Physics and Mathematics.
See the beginning of Chapter II.