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| Ernst Zermelo |
Ernst Zermelo (1871-1953) was a famous mathematician of the early
twentieth century. Among his achievements, the following can be mentioned:
- In 1899 he discovered Russell’s
paradox, two years before Russell. Although he did not
publish it, he did comment it with his colleagues at the University of
Göttingen, such as David Hilbert. Russell’s paradox proved that Cantor’s
set theory is inconsistent, since it makes it possible to build the
set of all sets that don’t belong to themselves. There are sets
that don’t belong to themselves, such as the set of even numbers, which is
not an even number. Others do belong to themselves, such as the set of
infinite sets, which is an infinite set. Now we can ask ourselves: Does
the set of all the sets that don’t belong to themselves belong to itself?
This question leads us to a paradox: if it does belong, it must belong;
and if it doesn’t belong, it mustn’t belong.
- In 1904 he proved the well-ordering theorem as the
first step towards proving the continuum hypothesis, the
first of Hilbert’s 23 unsolved problems. The well-ordering theorem states
that every set can be well-ordered, which means that any
non-empty ordered subset must have a minimum element. To prove it, he
proposed the axiom of choice,
which we will discuss later.
- In 1905 he began to work on an axiomatic set theory. His system,
improved in 1922 by Adolf Fraenkel, is a set of 8 axioms, which today is
called the Zermelo-Fraenkel
(ZF) system. Adding the axiom of choice to this system, we obtain the ZFC
system, which is most used today in set theory.
