text: Various schools of thought were opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which sought to ground mathematics on a small basis of a formal system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L.E.J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols. The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.
  Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable systemsuch as necessary to axiomatize the elementary theory of arithmetica statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth cannot be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This proves that there is no hope to prove the consistency of any system that contains an axiomatization of elementary arithmetic, and, in particular, to prove the consistency of Zermelo–Fraenkel set theory (ZFC), the system which is generally used for building all mathematics.