556 19. LOGIC AND SET THEORYmathematicians, since such proofs seldom give insight into the structures being
studied. For example, Euclid's proof that there are infinitely many primes proceeds
by assuming that the set of prime numbers is a finite set Ñ = {ñé,Ñ2é · • ·,Ñç} and
showing that in this case the number 1 + p\ • · • pn must either itself be a prime
number or be divisible by a prime different from ñ÷,... ,pn, which contradicts the
original assumption that pi,..., pn formed the entire set of prime numbers.
The appearance of starting with a false assumption and deriving a contradiction
can be avoided here by stating the theorem as follows: If there exists a set of ç
primes pi,..., pn, there exists a set of ç + 1 primes. The proof is exactly as before.
Nevertheless, the proof is still not intuitionistically valid, since there is no way of
saying whether or not 1 + p\ • • · p„ is prime.
In 1928 and 1929, a quarter-century after the debate over Zermelo's axiom of
choice, there was debate about intuitionism in the bulletin of the Belgian Royal
Academy of Sciences. Two Belgian mathematicians, M. Barzin and A. Errera, had
argued that Brouwer's logic amounted to a three-valued logic, since a statement
could be true, false, or undecidable. The opposite point of view was defended by
two distinguished Russian mathematicians, Aleksandr Yakovlevich Khinchin (1894-
1959) and Valerii Ivanovich Glivenko (1897-1940). Barzin and Errera had suggested
that to avoid three-valued logic, intuitionists ought to adopt as an axiom that if
ñ implies "q or r", then either ñ implies q or ñ implies r,^19 and also that if "p or
ò" implies r, then ñ implies r and q implies r. Starting from these principles of
Barzin and Errera and the trivial axiom "p or not-p" implies "p or nofc-p", Khinchin
deduced that ñ implies not-p and not-p implies p, thus reducing the suggestions of
Barzin and Errera to nonsense. Glivenko took only a little longer to show that,
in fact, Brouwer's logic was not three-valued. He proved that the statement "p or
not-p is false" is false in Brouwer's logic, and ultimately derived the theorem that
the statement "p is neither true nor false" is false (see Novosyolov, 2000).
A more "intuitive" objection to intuitionism is that intuition by its nature
cannot be codified as a set of rules. In adopting such rules, the intuitionists were
not being intuitionistic in the ordinary sense of the word. In any case, intuitionist
mathematics is obviously going to be somewhat sparser in results than mathematics
constructed on more liberal principles. That may be why it has attracted only a
limited group of adherents.
3.4. Mathematical practice. The paradoxes of naive set theory (such as Rus-
sell's paradox) were found to be avoidable if the word class is used loosely, as Cantor
had previously used the word set, but the word set is restricted to mean only a
class that is a member of some other class. (Classes that are not sets are called
proper classes.) Then to belong to a class A, a class  must not only fulfill the
requirements of the definition of the class A but must also be known in advance to
belong to some (possibly different) class.
This approach avoids Russell's paradox. The class C = {÷ : ÷ ö x} is ¢
class; its elements are those classes that belong to some class and are not elements
of themselves. If we now ask the question that led to Russell's paradox—Is C a
member of itself?—we do not reach a contradiction. If we assume C £ C, then
we conclude that C ö C, so that this assumption is not tenable. However, the
opposite assumption, that C £ C, is acceptable. It no longer leads to the conclusion
(^19) In the currently accepted semantics (metalanguage) of intuitionistic propositional calculus, if
"q or r" is a theorem, then either q is a theorem or r is a theorem.