§ 1. Set Theory
11
edifice, at least twenty years' work, and a part of his very self, by choosing
x tt x for the relation P{ x}. Suppose that there is indeed a set A such that
(2.1)
for every x, x E A is equivalent to x tt x.
Since a relation which is true for every x remains so when one substitutes
a specific mathematical object for the variable x, one sees that the relations
A E A and A t/:- A are logically equivalent: contradiction!
The axiom of separation (Ernest Zermelo, 1908) obviates "Russell's Paradox": if P{ x} is a proposition and X is a set one may speak of the set A of
those x which belong to X and satisfy P {x}; in logical language 15 :
(2.2)
(x E A) ~ (P{x} & (x EX));
instead of placing oneself in the absurd universe of all possible mathematical
objects one places oneself in the specific set X: this is one of the guard-rails of
the theory16. In particular, one may not speak of "the set of all sets", as was
done in Cantor's time, for if such a set X existed the relation x t/:- x would
define, by (2), a set A c X satisfying (1), an absurdity. You may certainly
think of the "class", "category", "totality" of sets, but this is not a set in the
technical sense of the term.
When X and Y are two sets one writes X - Y for the set of elements of X
that do not belong to Y: the axiom of separation legitimates this definition:
Pix} here is the relation x t/:- Y. By far the most frequent case is that when
Y eX; X - Y is then the complement of Y in X; in this case one obviously
has
X - (X - Y) = Y.
understand it, and makes no attempt to appear to understand it. This is a general
problem in the history of science: when it is written by scientists who understand
the subject the socio-political aspects disappear or reduce to non-documented
banalities, and vice versa. The exceptions, for instance Loup Verlet's book cited
above, are very rare. Dieudonne resolved this dilemma by saying that the sociopolitical aspects do not explain the scientists' ideas. Apart from the fact that
this statement may be false (obvious counterexamples: the logarithms of Napier
and Briggs for astronomers and navigators, Lavoisier and the French gunpowder
administration, Gauss and geodesy, Liebig and nitrogenous fertilisers, Haber and
the direct synthesis of ammonia, von Neumann after 1937, etc.), one is in the
right not to be solely interested in mathematics or physics in the strict sense.
15 The sign {=? denotes logical equivalence; the sign & indicates the conjunction
of two statements: the parentheses have the purpose of delimiting relations. See
the part of this chapter that treats mathematical logic.
16 The solution found by Zermelo trivially eliminates Russell's Paradox since then
one does not have the right to talk "in the air" of the set of x such that x rf. x:
one must specify at the outset that one places oneself in a given set. But the true
problem, resolved by Zermelo and his successors, was to show that on adopting
their axioms one did not forbid anything commonly done in mathematics. The
solution would have been rejected if, for example, it had made it impossible to
cunstruct the real numbers.
11
edifice, at least twenty years' work, and a part of his very self, by choosing
x tt x for the relation P{ x}. Suppose that there is indeed a set A such that
(2.1)
for every x, x E A is equivalent to x tt x.
Since a relation which is true for every x remains so when one substitutes
a specific mathematical object for the variable x, one sees that the relations
A E A and A t/:- A are logically equivalent: contradiction!
The axiom of separation (Ernest Zermelo, 1908) obviates "Russell's Paradox": if P{ x} is a proposition and X is a set one may speak of the set A of
those x which belong to X and satisfy P {x}; in logical language 15 :
(2.2)
(x E A) ~ (P{x} & (x EX));
instead of placing oneself in the absurd universe of all possible mathematical
objects one places oneself in the specific set X: this is one of the guard-rails of
the theory16. In particular, one may not speak of "the set of all sets", as was
done in Cantor's time, for if such a set X existed the relation x t/:- x would
define, by (2), a set A c X satisfying (1), an absurdity. You may certainly
think of the "class", "category", "totality" of sets, but this is not a set in the
technical sense of the term.
When X and Y are two sets one writes X - Y for the set of elements of X
that do not belong to Y: the axiom of separation legitimates this definition:
Pix} here is the relation x t/:- Y. By far the most frequent case is that when
Y eX; X - Y is then the complement of Y in X; in this case one obviously
has
X - (X - Y) = Y.
understand it, and makes no attempt to appear to understand it. This is a general
problem in the history of science: when it is written by scientists who understand
the subject the socio-political aspects disappear or reduce to non-documented
banalities, and vice versa. The exceptions, for instance Loup Verlet's book cited
above, are very rare. Dieudonne resolved this dilemma by saying that the sociopolitical aspects do not explain the scientists' ideas. Apart from the fact that
this statement may be false (obvious counterexamples: the logarithms of Napier
and Briggs for astronomers and navigators, Lavoisier and the French gunpowder
administration, Gauss and geodesy, Liebig and nitrogenous fertilisers, Haber and
the direct synthesis of ammonia, von Neumann after 1937, etc.), one is in the
right not to be solely interested in mathematics or physics in the strict sense.
15 The sign {=? denotes logical equivalence; the sign & indicates the conjunction
of two statements: the parentheses have the purpose of delimiting relations. See
the part of this chapter that treats mathematical logic.
16 The solution found by Zermelo trivially eliminates Russell's Paradox since then
one does not have the right to talk "in the air" of the set of x such that x rf. x:
one must specify at the outset that one places oneself in a given set. But the true
problem, resolved by Zermelo and his successors, was to show that on adopting
their axioms one did not forbid anything commonly done in mathematics. The
solution would have been rejected if, for example, it had made it impossible to
cunstruct the real numbers.