Mathematical theology

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.

Gödel and realism

Kurt Gödel

Kurt Gödel (1906-1978) was one of the most important mathematicians of the 20th century. In 1931, when he was 25, he rose to fame with his mathematical proof that the attempt to build a complete axiomatic system, from which one can deduce all the arithmetic of natural numbers or any equivalent system, is doomed to failure.

His first incompleteness theorem says the following:

Every consistent formal system as powerful as elementary arithmetic is not complete (it contains true undecidable propositions).

Let us look at a simplified informal demonstration:

Let theorem G say the following: This theorem G cannot be proved from the axioms and rules of system S.

• If we assume that Theorem G is false, system S is inconsistent, since a false theorem can be proved from the axioms and rules of S.
• Then if S is consistent, G must be true, and therefore cannot be proved from the axioms of S.
  

Gödel’s theorem shows that every axiomatic formalization of arithmetic is either inconsistent (it allows false theorems to be proved), or incomplete (it contains true theorems that cannot be proved).