§2. The logic of logicians
43
To construct the objects - sets - and relations which are the study of
mathematics, and not only logic, one needs, as we have seen, the three extra
symbols = [which the logicians tend to annex, subjecting it to the axiom
(Vx)(x = x))], E and 0, and a certain number of axioms, those set out in
the first part of this chapter; some permit the construction of new sets - of
unions, of pairs, of sets of sets, etc. - starting from given sets and relations;
others permit us to prove relations - equality, membership, inclusion, etc. -
between sets. As the use of logical and mathematical signs leads as well to
propositions as to sets, one needs to have a definition of sets, say, for example:
X is a set {::::::::} {(X = 0) or [:3x(x EX)]}.
To expound all this in strictly formalised language as the logicians do would
be pointless and unusable. Indeed, one learns the usage of set theory from
practice, and this requires only a little acquaintance with the subject.
It is now time to embark on true mathematics, where one has the agreeable
illusion of manipulating other things than boxes filled with emptiness, or
boxes filled with boxes filled with emptiness, or ... - of mathematics which
would never have interested anybody if not for this illusion and its surprising
appropriateness to "reality".